Answer
$f(x)=2x^{4}+5x^{3}+4x^{2}+5x+2$
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 $(i)$ is a zero, so is $(-i).$
$f(x)=a_{n}(x+2)(x+\displaystyle \frac{1}{2})(x-i)(x+i)$
$f(x)=a_{n}(x^{2}+\displaystyle \frac{5}{2}x+1)(x^{2}+1)$
Using the given $f(1),$we find $a_{n}:$
$f(1)=a_{n}(1+\displaystyle \frac{5}{2}+1)(1+1)$
$18=a_{n}(\displaystyle \frac{9}{2})(2)$
$a_{n}=2$
$f(x)=2(x^{2}+\displaystyle \frac{5}{2}x+1)(x^{2}+1)$
$f(x)=2(x^{4}+x^{2}+\displaystyle \frac{5}{2}x^{3}+\frac{5}{2}x+x^{2}+1)$
$f(x)=2(x^{4}+\displaystyle \frac{5}{2}x^{3}+2x^{2}+\frac{5}{2}x+1)$
$f(x)=2x^{4}+5x^{3}+4x^{2}+5x+2$
