College Algebra, Chapter 4, 4.4, Section 4.4, Problem 78

Determine all rational zeros of the polynomial $P(x) = 2x^4 + 15x^3 + 31x^2 + 20x + 4$ and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Rule of Signs, the quadratic formula or other factoring techniques.


So, $\displaystyle \frac{-1}{2}$ is a zero and $\displaystyle P(x) = \left( x + \frac{1}{2} \right) \left( 2x^3 + 14x^2 + 24x + 8 \right)$. We continue by factoring the quotient, the possible rational zeros of $P$ are $\displaystyle \pm \frac{1}{2}, \pm 1, \pm 2, \pm 4, \pm 8$ we check again the negative candidates first, beginning with the smallest.
Using Synthetic Division,


So $-2$ is a zero and $\displaystyle P(x) = \left(x + \frac{1}{2} \right) (x+2) (2x^2 + 10x + 4)$. We now factor the quotient using quadratic formula.

$
\begin{equation}
\begin{aligned}
x &= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\\
\\
x &= \frac{-10 \pm \sqrt{(10)^2 - 4(2)(4)}}{2(2)}\\
\\
x &= \frac{-5\pm\sqrt{17}}{2}
\end{aligned}
\end{equation}
$

Therefore, $\displaystyle P(x) = \left( x + \frac{1}{2} \right) (x + 2)\left( x + \frac{5+\sqrt{17}}{2}\right) \left( x + \frac{5-\sqrt{17}}{2}\right)$