Answer
$f(x)=3x^{3}+12x^{2}-93x-522$
Work Step by Step
The Linear Factorization Theorem... in words:
An nth-degree polynomial can be expressed as the product of a nonzero constant and $n$ linear factors, where each linear factor has a leading coefficient of 1.
Imaginary roots, if they exist, occur in conjugate pairs.
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If $(-5+2i)$ is a zero, so is $(-5-2i).$
$f(x)=a_{n}(x-6)(x+5+2i)(x+5-2i)$
$f(x)=a_{n}(x-6)(x^{2}+5x-2ix+5x+25-10i+2ix+10i-4i^{2})$
...$ i^{2}=-1$ ...
$f(x)=a_{n}(x-6)(x^{2}+10x+29)$
Using the given $f(2),$we find $a_{n}:$
$f(2)=a_{n}(2-6)(4+20+29)$
$-636=a_{n}(-4)(53)$
$-636=a_{n}(-212)$
$a_{n}=3$
$f(x)=3(x-6)(x^{2}+10x+29)$
$f(x)=3(x^{3}+10x^{2}+29x-6x^{2}-60x-174)$
$f(x)=3(x^{3}+4x^{2}-31x-174)$
$f(x)=3x^{3}+12x^{2}-93x-522$
