Question

h(x) = -x^4 + 3x^3 - 2x^2

Explanation:

Assuming the problem is to factor the function \( h(x) = -x^4 + 3x^3 - 2x^2 \), here's the step - by - step solution:

Step 1: Factor out the greatest common factor (GCF)

The GCF of the terms \(-x^4\), \(3x^3\), and \(-2x^2\) is \(-x^2\).
\[

$$\begin{align*} h(x)&=-x^4 + 3x^3 - 2x^2\\ &=-x^2(x^2 - 3x + 2) \end{align*}$$

\]

Step 2: Factor the quadratic expression

We need to factor the quadratic \(x^2 - 3x + 2\). We look for two numbers that multiply to \(2\) and add up to \(-3\). The numbers are \(-1\) and \(-2\).
\[

$$\begin{align*} x^2 - 3x + 2&=x^2 - x - 2x + 2\\ &=x(x - 1)-2(x - 1)\\ &=(x - 1)(x - 2) \end{align*}$$

\]

Step 3: Write the fully factored form

Substitute the factored quadratic back into the expression from Step 1.
\[
h(x)=-x^2(x - 1)(x - 2)
\]

Answer:

The factored form of \(h(x)=-x^4 + 3x^3 - 2x^2\) is \(\boldsymbol{-x^2(x - 1)(x - 2)}\)