Without actual division, prove that `2x^4-5x^3+2x^2-x+2` is exactly divisible by `x^2-3x+2.`

Explanation

To prove that the polynomial \( f(x) = 2x^4 - 5x^3 + 2x^2 - x + 2 \) is exactly divisible by \( g(x) = x^2 - 3x + 2 \) without actual division, we can use the Factor Theorem. According to the Factor Theorem, if \( g(x) \) is a factor of \( f(x) \), then \( f(a) = 0 \) for the roots \( a \) of \( g(x) \).

### Step 1: Factor \( g(x) \)

First, we factor the polynomial \( g(x) \):
\[ g(x) = x^2 - 3x + 2 \]
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