Answer
x = -1, -$\frac{1}{2}$, $\frac{1}{2}$, 1, 2.
Work Step by Step
The Rational Zero Test relies on finding possible zeros and testing to see if they are actually zeros. The possible zeros are all the factors of the constant term divided by all the factors of the leading coefficient.
f(x) = 4$x^{5}$ - 8$x^{4}$ - 5$x^{3}$ + 10$x^{2}$ + x - 2
Factors of the constant: $\pm$ 1, $\pm$2
Factors of the leading coefficient: $\pm$1, $\pm$2, $\pm$4
All possible combinations:
x = $\frac{\pm 1}{\pm 1}$, $\frac{\pm 2}{\pm 1}$, $\frac{\pm 1}{\pm 4}$, $\frac{\pm 2}{\pm 1}$, $\frac{\pm 2}{\pm 2}$, $\frac{\pm 2}{\pm 4}$
x = 1, -1, 2, -2, $\frac{1}{4}$, -$\frac{1}{4}$, $\frac{1}{2}$, -$\frac{1}{2}$
The line in the graph passes through the x-axis at x = -1, -$\frac{1}{2}$, $\frac{1}{2}$, 1, 2.
Since those are all possible zeros they are confirmed by the graph.
