To determine the height of the prism, we use the formula for the volume of a prism:

$\text{Volume} = \text{Base Area} \times \text{Height}$

Given:

$\text{Volume} = 180x^3 - 150x^2 + 20x$ $\text{Area of Base} = 18x^2 - 15x + 2$

We can find the height by rearranging the formula:

$\text{Height} = \frac{\text{Volume}}{\text{Area of Base}}$

This involves dividing $180x^3 - 150x^2 + 20x$ by $18x^2 - 15x + 2$. Let’s perform this polynomial division:

1. Divide the leading terms: $\frac{180x^3}{18x^2} = 10x$.
2. Multiply $10x$ by $18x^2 - 15x + 2$ and subtract from the original volume polynomial.
3. Repeat the process with the new polynomial obtained after subtraction.

Would you like me to show the full division process?

Questions for further exploration:

1. How is polynomial division performed?
2. What are the applications of dividing polynomials in geometry?
3. How can we verify if our division result is correct?
4. Can this process be used for other types of shapes?
5. How would the process differ if the area of the base was not a polynomial?

Tip: Polynomial division is similar to long division with numbers; match terms with the highest degree first.