To simplify the expression \(\frac{8x^3 - 12x^2 + 18x - 27}{2x - 3} - x^2 + 4x\), we will first perform polynomial long division on the first part, \(\frac{8x^3 - 12x^2 + 18x - 27}{2x - 3}\).
Step 1: Perform Polynomial Long Division
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Divide the leading term: Divide \(8x^3\) by \(2x\) which gives \(4x^2\).
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Multiply: \(4x^2(2x - 3) = 8x^3 - 12x^2\).
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Subtract: \[ (8x^3 - 12x^2 + 18x - 27) - (8x^3 - 12x^2) = 18x - 27. \]
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Bring down: Now, use \(2x - 3\) to divide the resulting polynomial \(18x - 27\).
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Divide the leading term: Divide \(18x\) by \(2x\) which gives \(9\).
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Multiply: \(9(2x - 3) = 18x - 27\).
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Subtract: \[ (18x - 27) - (18x - 27) = 0. \]
Since the remainder is 0, we conclude that: \[ \frac{8x^3 - 12x^2 + 18x - 27}{2x - 3} = 4x^2 + 9. \]
Step 2: Substitute Back Into the Original Expression
Now we substitute back into the expression: \[ 4x^2 + 9 - x^2 + 4x. \]
Step 3: Combine Like Terms
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Combine \(4x^2 - x^2\): \[ 4x^2 - x^2 = 3x^2. \]
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Bring down the \(4x\) and the \(9\): \[ 3x^2 + 4x + 9. \]
Final Result
Thus, the simplified expression is: \[ \boxed{3x^2 + 4x + 9}. \]
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