Solution

\[ \begin{align*} \color{green} (a+b+c)^2 &= \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\[4pt] [(x)+(-2y)+(3z)]^2 & = (x)^2 + (-2y)^2 + (3z)^2 + 2 (x) (-2y) + 2 (-2y) (3z) + 2 (3z) (x) \\[4pt] & = \color{red} x^2 + 4y^2 + 9z^2 - 4xy - 12yz + 6xz \end{align*} \]

Solution

\[ \begin{align*} \color{green} (a+b+c)^2 &= \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\[4pt] [(-5x)+(2y)+(z)]^2 & = (-5x)^2 + (2y)^2 + (z)^2 + 2 (-5x) (2y) + 2 (2y) (z) + 2 (z) (-5x) \\[4pt] & = \color{red} 25x^2 + 4y^2 + z^2 - 20xy + 4yz - 10xz \end{align*} \]

Solution

\[ \begin{align*} \color{green} (a+b+c)^2 &= \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\[4pt] [(4x)+(y)+(-3z)]^2 & = (4x)^2 + (y)^2 + (-3z)^2 + 2 (4x)(y) + 2 (y) (-3z) + 2 (-3z) (4x) \\[4pt] & = \color{red} 16x^2 + y^2 + 9z^2 + 8xy - 6yz - 24xz \end{align*} \]

Solution

\[ \begin{align*} \color{green} (a+b+c)^2 &= \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\[4pt] [(3a)+(-5b)+(-7c)]^2 & = (3a)^2 + (-5b)^2 + (-7c)^2 + 2 (3a) (-5b) + 2 (-5b) (-7c) + 2 (-7c) (3a) \\[4pt] & = \color{red} 9a^2 + 25b^2 + 49c^2 - 30ab + 70bc - 42ac \end{align*} \]

Solution

\[ \begin{align*} \color{green} (a+b+c)^2 &= \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\[4pt] [(-2a)+(-b)+(3c)]^2 & = (-2a)^2 + (-b)^2 + (3c)^2 + 2 (-2a)(-b) + 2 (-b)(3c) + 2(3c)(-2a) \\[4pt] & = \color{red} 4a^2 + b^2 + 9c^2 + 4ab - 6bc - 12ac \end{align*} \]

Solution

\[ \begin{align*} \color{green} (a+b+c)^2 &= \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\[4pt] [(-a)+(6b)+(-2c)]^2 & = (-a)^2 + (6b)^2 + (-2c)^2 + 2 (-a) (6b) + 2 (6b) (-2c) + 2 (-2c)(-a) \\[4pt] & = \color{red} a^2 + 36b^2 + 4c^2 - 12ab - 24bc + 4ac \end{align*} \]

Solution

\[ \begin{align*} \color{green} (a+b+c)^2 &= \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\[4pt] [(1)+(2x)+(-3y)]^2 & = (1)^2 + (2x)^2 + (-3y)^2 + 2 (1) (2x) + 2 (2x) (-3y) + 2 (-3y) (1) \\[4pt] & = \color{red} 1 + 4x^2 + 9y^2 + 4x - 12xy - 6y \end{align*} \]

Solution

\[ \begin{align*} \color{green} (a+b+c)^2 &= \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\[4pt] [(2x)+(-4y)+(-1)]^2 & = (2x)^2 + (-4y)^2 + (-1)^2 + 2 (2x) (-4y) + 2 (-4y)(-1) + 2 (-1) (2x) \\[4pt] & = \color{red} 4x^2 + 16y^2 + 1 - 16xy + 8y - 4x \end{align*} \]

Solution

\[ \begin{align*} \color{green} (a+b+c)^2 &= \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\[4pt] [(6p)+(-5q)+(-4r)]^2 & = (6p)^2 + (-5q)^2 + (-4r)^2 + 2 (6p) (-5q) + 2(-5q)(-4r) + 2 (-4r)(6p) \\[4pt] & = \color{red} 36p^2 + 25q^2 + 16r^2 - 60pq + 40qr - 48pr \end{align*} \]

Solution

\[ \begin{align*} \color{green} (a+b+c)^2 &= \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\[4pt] (p + 5q + 2)^2 & = (p)^2 + (5q)^2 + (2)^2 + 2 (p) (5q) + 2 (5q) (2) + 2 (2) (p) \\[4pt] & = \color{red} p^2 + 25q^2 + 4 + 10pq + 20q + 4p \end{align*} \]

Solution

\[ \begin{align*} \color{green} (a+b+c)^2 &= \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\[4pt] [(3x)+(-4y)+(-5)]^2 & = (3x)^2 + (-4y)^2 + (-5)^2 + 2 (3x) (-4y) + 2 (-4y) (-5) + 2 (-5) (3x) \\[4pt] & = \color{red} 9x^2 + 16y^2 + 25 - 24xy + 40y - 30x \end{align*} \]

Solution

\[ \begin{align*} \color{green} (a+b+c)^2 &= \color{green} a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\[4pt] [(-2x)+(6y)+(4)]^2 & = (-2x)^2 + (6y)^2 + (4)^2 + 2 (-2x) (6y) + 2 (6y) (4) + 2 (4)(-2x) \\[4pt] & = \color{red} 4x^2 + 36y^2 + 16 - 24xy + 48y - 16x \end{align*} \]