Solution

\[ \begin{aligned} \text{Dividend} &= \color{magenta}{x^3 + 0x^2 + (k+8)x + k} \\[4pt] \text{Divisor} &= \color{magenta}{x - 2} \end{aligned} \] \[ \begin{array}{l} \hspace{1.6cm} x^2 \ + \ 2x \ + \ (k+12) \\ x-2 \enclose{longdiv}{\ \ x^3 + 0x^2 + (k+8)x + k}\\ \hspace{1cm}\overset{\color{red}\scriptsize(-)}{+}x^3 \overset{\color{red}\scriptsize(+)}{-}2x^2 \\ \hline \hspace{1.5cm}0 + 2x^2 + (k+8)x + k \\ \hspace{1.9cm}\overset{\color{red}\scriptsize(-)}{+}2x^2 \ \ \overset{\color{red}\scriptsize(+)}{-}4x \\ \hline \hspace{2.8cm}0 + (k+12)x + k \\ \hspace{3.2cm}\overset{\color{red}\scriptsize(-)}{+}(k+12)x \overset{\color{red}\scriptsize(+)}{-} 2(k+12) \\ \hline \hspace{6cm} 3k+24 \\ \hline \end{array} \]\[ \begin{aligned} 1^{st} \text{ Remainder} &= \color{green} 3k+24 \\ \\ \text{Dividend} &= \color{magenta}{x^3 + 0x^2 + (k+8)x + k} \\[4pt] \text{Divisor} &= \color{magenta}{x + 1} \end{aligned} \] \[ \begin{array}{l} \hspace{1.6cm} x^2 \ - \ x \ + \ (k+9) \\ x+1 \enclose{longdiv}{\ \ x^3 + 0x^2 + (k+8)x + k}\\ \hspace{1.1cm}\overset{\color{red}\scriptsize(-)}{+}x^3 \overset{\color{red}\scriptsize(-)}{+}x^2 \\ \hline \hspace{1.6cm}0 - x^2 + (k+8)x + k \\ \hspace{2cm}\overset{\color{red}\scriptsize(+)}{-}x^2 \hspace{.7cm} \overset{\color{red}\scriptsize(+)}{-}x \\ \hline \hspace{2.6cm}0 + (k+9)x + k \\ \hspace{2.9cm}\overset{\color{red}\scriptsize(-)}{+}(k+9)x \overset{\color{red}\scriptsize(-)}{+} (k+9) \\ \hline \hspace{4cm} 0 \hspace{1cm} - 9 \\ \hline \end{array} \] \[ \begin{aligned} 2^{nd} \text{ Remainder} &= \color{green} -9 \\ \\ 1^{st} \text{ Remainder} + 2^{nd}\text{ Remainder} &= 0 \\[4pt] (3k+24) + (-9) &= 0 \\[4pt] 3k+24 - 9 &= 0 \\[4pt] 3k + 15 &= 0 \\[4pt] \color{green}{k} &= \color{green}{-5} \end{aligned} \]

Answer \(\color{red}{k=-5}\)