The division problem presented in the image is:

$\frac{-x^5 + 12x^4 - 18x^3 + 12}{2x^3 - 4x^2 + 2}$

Let me perform the polynomial division and simplify the expression.

I'll solve this step-by-step.The result of the polynomial division is:

$\frac{-x^5 + 12x^4 - 18x^3 + 12}{2x^3 - 4x^2 + 2} = -\frac{1}{2}x^2 + 5x + 1 + \frac{5x^2 - 10x + 10}{2x^3 - 4x^2 + 2}$

This means the quotient is $-\frac{1}{2}x^2 + 5x + 1$, and the remainder is $\frac{5x^2 - 10x + 10}{2x^3 - 4x^2 + 2}$.

Would you like a further breakdown of the division steps or explanation of the remainder term?

Here are 5 related questions:

1. How does polynomial long division work in more detail?
2. What happens when the remainder is not zero in polynomial division?
3. How can we simplify the remainder in this division?
4. What real-life applications use polynomial division?
5. What are the conditions for a polynomial division to result in a clean quotient (without remainder)?

Tip: Always check if you can factor the denominator and numerator before division; this might simplify the process!