Answer
$x\in \{2\}$
Work Step by Step
See The Rational Zero Theorem:
... If $\displaystyle \frac{p}{q}$ is a zero of the polynomial $f(x) $with integer coefficients,
then $p$ is a factor of the constant term, $a_{0}$, and
$q$ is a factor of the leading coefficient, $a_{n}$.
------------------------
$f(x)=x^4-4x^3+9x^2-20x+20$
a. candidates for zeros, $\displaystyle \frac{p}{q}:$
$p:\qquad \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20$
$q:\qquad \pm 1, $
$\displaystyle \frac{p}{q}:\qquad \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20$
b. Try for $x=2:$
$\begin{array}{lllll}
\underline{2}| &1& -4 & 9 & -20 & 20\\
& & 2 & -4 & 10&-20\\
& -- & -- & -- & --\\
& 1 & -2 & 5&-10 & |\underline{0}
\end{array}$
$2$ is a zero,
$f(x)=(x-2)(x^3-2x^2+5x-10)$
Try for $x=2:$
$\begin{array}{lllll}
\underline{2}| & 1 & -2 & 5 & -10\\
& & 2 & 0&10\\
& -- & -- & -- & --\\
& 1 & 0&5 & |\underline{0}
\end{array}$
$2$ is a zero,
$f(x)=(x-2)^2(x^2+5)$
$x\in \{2\}$
