Solution

\[ \begin{align*} \color{green} (x + a)(x + b) &= \color{green} x^2 + (a+b)x + ab \\[4pt] (x + 5)(x + 4) & = x^2 + (5+4)x + (5 \times 4) \\[4pt] & = \color{red} x^2 + 9x + 20 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x + a)(x + b) &= \color{green} x^2 + (a+b)x + ab \\[4pt] (a + 3)(a + 6) & = a^2 + (3+6)a + (3 \times 6) \\[4pt] & = \color{red} a^2 + 9a + 18 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x - a)(x + b) &= \color{green} x^2 + (b-a)x - ab \\[4pt] (x - 9)(x + 7) & = x^2 + (7-9)x - (9 \times 7) \\[4pt] & = x^2 + (-2)x -63 \\[4pt] & = \color{red} x^2 - 2x -63 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x + a)(x - b) &= \color{green} x^2 + (a-b)x - ab \\[4pt] (x + 8)(x - 5) & = x^2 + (8 - 5)x - (8 \times 5) \\[4pt] & = \color{red} x^2 + 3x - 40 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x - a)(x - b) &= \color{green} x^2 - (a+b)x + ab \\[4pt] (z - 3)(z - 1) & = z^2 - (3+1)z + (3 \times 1) \\[4pt] & = \color{red} z^2 - 4z + 3 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x - a)(x - b) &= \color{green} x^2 - (a+b)x + ab \\[4pt] (p - 5)(p - 4) & = p^2 - (5+4)p + (5 \times 4) \\[4pt] & = \color{red} p^2 - 9p + 20 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x - a)(x + b) &= \color{green} x^2 + (b-a)x - ab \\[4pt] (y - 1)(y + 2) & = y^2 + (2-1)y - (1 \times 2) \\[4pt] & = \color{red} y^2 + y - 2 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x + a)(x - b) &= \color{green} x^2 + (a-b)x - ab \\[4pt] (z + 3)(z - 7) & = z^2 + (3 - 7)z - (3 \times 7) \\[4pt] & = z^2 +(- 4)z - 21 \\[4pt] & = \color{red} z^2 - 4z - 21 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x + a)(x - b) &= \color{green} x^2 + (a-b)x - ab \\[4pt] (p + 8)(p - 3) & = p^2 + (8-3)p - (8 \times 3) \\[4pt] & = \color{red} p^2 + 5p - 24 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x + a)(x - b) &= \color{green} x^2 + (a-b)x - ab \\[4pt] (z + 6)(z - 5) & = z^2 + (6 - 5)z - (6 \times 5) \\[4pt] & = \color{red} z^2 + z - 30 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x - a)(x - b) &= \color{green} x^2 - (a+b)x + ab \\[4pt] (x - 6)(x - 9) & = x^2 - (6+9)x + (6 \times 9) \\[4pt] & = \color{red} x^2 - 15x + 54 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x - a)(x + b) &= \color{green} x^2 + (b-a)x - ab \\[4pt] (x - 10)(x + 9) & = x^2 + (9 - 10)x - (10 \times 9) \\[4pt] & = x^2 +(- 1)x - 90 \\[4pt] & = \color{red} x^2 - x - 90 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (a - b)(a + b) &= \color{green} a^2 - b^2 \\[4pt] (y - 4)(y + 4) & = y^2 - (4)^2 \\[4pt] & = \color{red} y^2 - 16 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x - a)(x - b) &= \color{green} x^2 - (a+b)x + ab \\[4pt] (x - 4)(x - 14) & = x^2 - (4+14)x + (4 \times 14) \\[4pt] & = \color{red} x^2 - 18x + 56 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x - a)(x - b) &= \color{green} x^2 - (a+b)x + ab \\[4pt] (x - 8)(x - 2) & = x^2 - (8+2)x + (8 \times 2) \\[4pt] & = \color{red} x^2 - 10x + 16 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x + a)(x + b) &= \color{green} x^2 + (a+b)x + ab \\[4pt] 102 \times 104 & = (100 + 2)(100 + 4) \\[4pt] & = (100)^2 + (2 + 4) \times 100 + (2 \times 4) \\[4pt] & = 10000 + 6 \times 100 + 8 \\[4pt] & = 10000 + 600 + 8 \\[4pt] & = 10600 + 8 \\[4pt] & = \color{red} 10608 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x + a)(x + b) &= \color{green} x^2 + (a+b)x + ab \\[4pt] 105 \times 103 & = (100 + 5)(100 + 3) \\[4pt] & = (100)^2 + (5 + 3) \times 100 + (5 \times 3) \\[4pt] & = 10000 + 8 \times 100 + 15 \\[4pt] & = 10000 + 800 + 15 \\[4pt] & = 10800 + 15 \\[4pt] & = \color{red} 10815 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x + a)(x + b) &= \color{green} x^2 + (a+b)x + ab \\[4pt] 206 \times 205 & = (200 + 6)(200 + 5) \\[4pt] & = (200)^2 + (6 + 5) \times 200 + (6 \times 5) \\[4pt] & = 40000 + 11 \times 200 + 30 \\[4pt] & = 40000 + 2200 + 30 \\[4pt] & = 42200 + 30 \\[4pt] & = \color{red} 42230 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x - a)(x - b) &= \color{green} x^2 - (a+b)x + ab \\[4pt] 98 \times 96 & = (100 - 2)(100 - 4) \\[4pt] & = (100)^2 - (2+4) \times 100 + (2 \times 4) \\[4pt] & = 10000 - 6 \times 100 + 8 \\[4pt] & = 10000 - 600 + 8 \\[4pt] & = 10008 - 600 \\[4pt] & = \color{red} 9408 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x + a)(x + b) &= \color{green} x^2 + (a+b)x + ab \\[4pt] 87 \times 85 & = (80 +7)(80 + 5) \\[4pt] & = (80)^2 + (7+5) \times 80 + (7 \times 5) \\[4pt] & = 6400 + 12 \times 80 + 35 \\[4pt] & = 6400 + 960 + 35 \\[4pt] & = 7360 + 35 \\[4pt] & = \color{red} 7395 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x + a)(x - b) &= \color{green} x^2 + (a-b)x - ab \\[4pt] 104 \times 95 & = (100 + 4)(100 - 5) \\[4pt] & = (100)^2 + (4 - 5) \times 100 - (4 \times 5) \\[4pt] & = 10000 - 1 \times 100 - 20 \\[4pt] & = 10000 - 100 - 20 \\[4pt] & = 10000 - 120 \\[4pt] & = \color{red} 9880 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x - a)(x + b) &= \color{green} x^2 + (b-a)x - ab \\[4pt] 97 \times 102 & = (100 - 3)(100 + 2) \\[4pt] & = (100)^2 + (2-3) \times 100 - (3 \times 2) \\[4pt] & = 10000 - 1 \times 100 - 6 \\[4pt] & = 10000 - 100 - 6 \\[4pt] & = 10000 - 106 \\[4pt] & = \color{red} 9894 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x + a)(x - b) &= \color{green} x^2 + (a-b)x - ab \\[4pt] 203 \times 198 & = (200 + 3)(200 - 2) \\[4pt] & = (200)^2 + (3 - 2) \times 200 - (3 \times 2) \\[4pt] & = 40000 + 1 \times 200 - 6 \\[4pt] & = 40000 + 200 - 6 \\[4pt] & = 40200 - 6 \\[4pt] & = \color{red} 40194 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x + a)(x + b) &= \color{green} x^2 + (a+b)x + ab \\[4pt] 35 \times 37 & = (30+5)(30 + 7) \\[4pt] & = (30)^2 + (5+7) \times 30 + (5 \times 7) \\[4pt] & = 900 +12 \times 30 +35 \\[4pt] & = 900 + 360 + 35 \\[4pt] & = 1260 + 35 \\[4pt] & = \color{red} 1295 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x + a)(x - b) &= \color{green} x^2 + (a-b)x - ab \\[4pt] 106 \times 93 & = (100 + 6)(100 - 7) \\[4pt] & = (100)^2 + (6-7) \times 100 - (6 \times 7) \\[4pt] & = 10000 - 1 \times 100 - 42 \\[4pt] & = 10000 - 100 - 42 \\[4pt] & = 10000 - 142 \\[4pt] & = \color{red} 9858 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x + a)(x + b) &= \color{green} x^2 + (a+b)x + ab \\[4pt] (x^2 + 3)(x^2 + 4) & = (x^2)^2 + (3+4)x^2 + (3 \times 4) \\[4pt] & = \color{red} x^4 + 7x^2 + 12 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x + a)(x + b) &= \color{green} x^2 + (a+b)x + ab \\[6pt] \left( x + \dfrac{4}{3} \right) \left( x + \dfrac{1}{3} \right) & = x^2 + \left(\frac{4}{3} + \frac{1}{3}\right)x + \frac{4}{3} \times \frac{1}{3} \\[6pt] & = \color{red} x^2 + \frac{5}{3}x + \frac{4}{9} \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x - a)(x - b) &= \color{green} x^2 - (a+b)x + ab \\[6pt] \left( x - \dfrac{3}{5} \right) \left( x - \dfrac{1}{2} \right) & = (x)^2 - \left(\frac{3}{5} + \frac{1}{2}\right)x + \left(\frac{3}{5} \times \frac{1}{2}\right) \\[6pt] & = x^2 - \left(\frac{6+5}{10} \right)x + \frac{3}{10} \\[6pt] & = \color{red} x^2 - \frac{11}{10}x + \frac{3}{10} \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x - a)(x + b) &= \color{green} x^2 + (b-a)x - ab \\[4pt] (y^2 - 6)(y^2 + 7) & = (y^2)^2 + \left(7-6\right)y^2 - (6 \times 7) \\[4pt] & = y^4 + (1)y^2 - 42 \\[4pt] & = \color{red} y^4 + y^2 - 42 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x + a)(x - b) &= \color{green} x^2 + (a-b)x - ab \\[6pt] \left( z^2 + 4 \right) \left( z^2 - \dfrac{1}{4} \right) & = (z^2)^2 + \left(4 -\frac{1}{4} \right)z^2 - \left({\cancel4}^1 \times \frac{1}{{\cancel4}_1}\right) \\[6pt] & = z^4 + \left( \frac{16 - 1}{4}\right)z^2 -1 \\[6pt] & = \color{red} z^4 + \frac{15}{4}z^2 - 1 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x - a)(x - b) &= \color{green} x^2 - (a+b)x + ab \\[4pt] (y^2 - 3)(y^2 - 1) & = (y^2)^2 - (3+1)y^2 + (3 \times 1) \\[4pt] & = \color{red} y^4 - 4y^2 + 3 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x + a)(x + b) &= \color{green} x^2 + (a+b)x + ab \\[4pt] (x^3 + 5)(x^3 + 2) & = (x^3)^2 + (5 + 2)x^3 + (5 \times 2) \\[4pt] & = \color{red} x^6 + 7x^3 + 10 \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x - a)(x + b) &= \color{green} x^2 + (b-a)x - ab \\[6pt] \left( p^2 - \frac{1}{4} \right) \left( p^2 + \frac{1}{8} \right) & = (p^2)^2 + \left(\frac{1}{8} -\frac{1}{4} \right)p^2 - \left(\frac{1}{4} \times \frac{1}{8}\right) \\[6pt] & = p^4 + \left(\frac{1-2}{8} \right)p^2 - \frac{1}{32} \\[6pt] & = p^4 + \left(\frac{-1}{8} \right)p^2 - \frac{1}{32} \\[6pt] & = \color{red} p^4 - \frac{1}{8}p^2 - \frac{1}{32} \end{align*} \]

Solution

\[ \begin{align*} \color{green} (x + a)(x + b) &= \color{green} x^2 + (a+b)x + ab \\[6pt] \left( z + \frac{1}{6} \right)(z + 6) & = (z)^2 + \left(\frac{1}{6} + 6\right)z + \left(\frac{1}{{\cancel6}_1} \times {\cancel6}^1\right) \\[6pt] & = z^2 + \left(\frac{1+36}{6} \right)z + 1 \\[6pt] & = \color{red} z^2 + \frac{37}{6}z + 1 \end{align*} \]