To perform long division on the polynomials \( 24x^4 - 24x^3 - 18x^2 \) divided by \( 4x^3 + 2x^2 \), we will go through the steps as follows:
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Set up the long division: Write \( 24x^4 - 24x^3 - 18x^2 \) under the division symbol and \( 4x^3 + 2x^2 \) outside.
\[ \begin{array}{r|l} 4x^3 + 2x^2 & 24x^4 - 24x^3 - 18x^2 \ \end{array} \]
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Divide the leading term: Divide the leading term of the dividend \( 24x^4 \) by the leading term of the divisor \( 4x^3 \):
\[ \frac{24x^4}{4x^3} = 6x \]
Write \( 6x \) above the division line.
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Multiply back: Multiply \( 6x \) by the entire divisor \( (4x^3 + 2x^2) \):
\[ 6x(4x^3 + 2x^2) = 24x^4 + 12x^3 \]
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Subtract: Subtract \( 24x^4 + 12x^3 \) from the original polynomial:
\[ (24x^4 - 24x^3 - 18x^2) - (24x^4 + 12x^3) = -24x^3 - 12x^3 - 18x^2 = -36x^3 - 18x^2 \]
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Bring down the next term: There are no more terms to bring down.
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Repeat the process: Now, we divide the leading term of the new polynomial \( -36x^3 \) by the leading term of the divisor \( 4x^3 \):
\[ \frac{-36x^3}{4x^3} = -9 \]
Write \(-9\) above the division line next to \(6x\).
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Multiply back: Multiply \(-9\) by the entire divisor \( (4x^3 + 2x^2) \):
\[ -9(4x^3 + 2x^2) = -36x^3 - 18x^2 \]
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Subtract again: Subtract \(-36x^3 - 18x^2\) from \(-36x^3 - 18x^2\):
\[ (-36x^3 - 18x^2) - (-36x^3 - 18x^2) = 0 \]
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Conclusion: Since the remainder is \(0\), the division is exact, and our quotient is:
\[ 6x - 9 \]
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Final answer in standard form: The result of the polynomial long division is:
\[ \boxed{6x - 9} \]