Use polynomial long division to divide

-4x3 + 24x2 – 24x + 4
by
x2 – 5x + 1

..it's difficult to show long division here,, but the answer is 4 - 4x.

To divide -4x^3 + 24x^2 – 24x + 4 by x^2 – 5x + 1 using polynomial long division, follow these steps:

1. Arrange both the dividend and divisor in standard form, meaning descending order of exponents.
-4x^3 + 24x^2 – 24x + 4 (dividend)
x^2 – 5x + 1 (divisor)

2. Divide the first term of the dividend by the first term of the divisor. Place the quotient above the line.
-4x^3 / x^2 = -4x

3. Multiply the entire divisor by the quotient obtained in step 2.
-4x * (x^2 – 5x + 1) = -4x^3 + 20x^2 - 4x

4. Subtract the result obtained in step 3 from the dividend.
-4x^3 + 24x^2 – 24x + 4 - (-4x^3 + 20x^2 - 4x) = 4x^2 - 20x + 4

5. Bring down the next term of the dividend.
4x^2 - 20x + 4

6. Repeat steps 2-5 until all terms have been processed.

4x^2 / x^2 = 4
4 * (x^2 – 5x + 1) = 4x^2 - 20x + 4
(4x^2 - 20x + 4) - (4x^2 - 20x + 4) = 0 (no remainder)

The quotient is -4x + 4, and there is no remainder.

Therefore, -4x^3 + 24x^2 – 24x + 4 divided by x^2 – 5x + 1 equals -4x + 4.