To perform the division of the polynomial \( (24x^4 + 18x^3 - 9x^2 + 3) \) by \( (4x^2 - 3x + 1) \), we will use polynomial long division.
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Set up the long division: We place \( 24x^4 + 18x^3 - 9x^2 + 3 \) under the long division symbol and \( 4x^2 - 3x + 1 \) on the outside.
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Divide the leading terms: Take the leading term of the numerator \( 24x^4 \) and divide it by the leading term of the denominator \( 4x^2 \):
\[ 24x^4 \div 4x^2 = 6x^2 \]
Write \( 6x^2 \) above the division line.
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Multiply and subtract: Multiply \( 6x^2 \) by \( 4x^2 - 3x + 1 \):
\[ 6x^2(4x^2 - 3x + 1) = 24x^4 - 18x^3 + 6x^2 \]
Subtract this from the original polynomial:
\[ (24x^4 + 18x^3 - 9x^2 + 3) - (24x^4 - 18x^3 + 6x^2) = 36x^3 - 15x^2 + 3 \]
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Repeat the process: Now, take the new leading term \( 36x^3 \) and divide by \( 4x^2 \):
\[ 36x^3 \div 4x^2 = 9x \]
Write \( 9x \) above the line next to \( 6x^2 \).
Now multiply \( 9x \) by \( 4x^2 - 3x + 1 \):
\[ 9x(4x^2 - 3x + 1) = 36x^3 - 27x^2 + 9x \]
Subtract this from \( 36x^3 - 15x^2 + 3 \):
\[ (36x^3 - 15x^2 + 3) - (36x^3 - 27x^2 + 9x) = 12x^2 - 9x + 3 \]
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Repeat again: Divide \( 12x^2 \) by \( 4x^2 \):
\[ 12x^2 \div 4x^2 = 3 \]
Write \( 3 \) above the line next to \( 9x \).
Multiply \( 3 \) by \( 4x^2 - 3x + 1 \):
\[ 3(4x^2 - 3x + 1) = 12x^2 - 9x + 3 \]
Subtract this from \( 12x^2 - 9x + 3 \):
\[ (12x^2 - 9x + 3) - (12x^2 - 9x + 3) = 0 \]
Since we are left with a remainder of 0, the final quotient from the division is:
\[ \text{Quotient} = 6x^2 + 9x + 3 \]
Thus, the quotient of \( (24x^4 + 18x^3 - 9x^2 + 3) \div (4x^2 - 3x + 1) \) is:
\[ \boxed{6x^2 + 9x + 3} \]