Express $$ f(x) $$ in the form $$ f(x)=(x-k) q(x)+r $$ for the given value of $$ k $$ $$ f(x)=x^{3}+5 x^{2}+10 x+8, k=-2 $$ $$ f(x)=\square $$

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To express $$ f(x) $$ in the form $$ f(x)=(x-k) q(x)+r $$ for the given value of $$ k $$, we need to perform polynomial long division or synthetic division to divide $$ f(x) $$ by $$ x-k $$.

Given:
$$ f(x) = x^3 + 5x^2 + 10x + 8 $$
$$ k = -2 $$

We want to divide $$ f(x) $$ by $$ x + 2 $$ (since $$ k = -2 $$ and we're looking for $$ (x - k) $$).

Let's perform the division:

$$
\begin{array}{c|ccccc}
\multicolumn{2}{r}{x^2} & +3x & +4 \
\cline{2-6}
x + 2 & x^3 & +5x^2 & +10x & +8 \
\multicolumn{2}{r}{- (x^3 & +2x^2)} & & & \
\cline{2-3}
\multicolumn{2}{r}{0 & 3x^2 & +10x} & & \
\multicolumn{2}{r}{- (3x^2 & +6x)} & & & \
\cline{2-3}
\multicolumn{2}{r}{0 & 4x & +8} & & \
\multicolumn{2}{r}{- (4x & +8)} & & & \
\cline{2-3}
\multicolumn{2}{r}{0 & 0 & 0} & & \
\end{array}
$$

The quotient $$ q(x) $$ is $$ x^2 + 3x + 4 $$, and the remainder $$ r $$ is $$ 0 $$.

Therefore, we can express $$ f(x) $$ as:

$$ f(x) = (x + 2)(x^2 + 3x + 4) + 0 $$

Since the remainder is $$ 0 $$, the expression simplifies to:

$$ f(x) = (x + 2)(x^2 + 3x + 4) $$

So, the final answer is:

$$ f(x) = (x + 2)(x^2 + 3x + 4) $$
$$ f(x) = (x + 2)(x^2 + 3x + 4) $$

Express in the form for the given value of

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