Question

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Explanation:

Step1: Divide each term by \(-4x^2\)

We have the polynomial \(-8x^5 + 32x^4 + 24x^3 + 12x^2\) and we divide each term by \(-4x^2\).

For the first term: \(\frac{-8x^5}{-4x^2}\)
Using the rule of exponents \(\frac{a^m}{a^n}=a^{m - n}\) and \(\frac{-8}{-4}=2\), we get \(2x^{5-2}=2x^3\)

For the second term: \(\frac{32x^4}{-4x^2}\)
\(\frac{32}{-4}=-8\) and \(x^{4 - 2}=x^2\), so we have \(-8x^2\)

For the third term: \(\frac{24x^3}{-4x^2}\)
\(\frac{24}{-4}=-6\) and \(x^{3-2}=x\), so we get \(-6x\)

For the fourth term: \(\frac{12x^2}{-4x^2}\)
\(\frac{12}{-4}=-3\) and \(x^{2-2}=x^0 = 1\), so we have \(-3\)

Step2: Combine the results

Combining all the terms we got from each division: \(2x^3-8x^2 - 6x-3\)

Answer:

\(2x^3 - 8x^2 - 6x - 3\)