To perform the operation and simplify the expression \( \frac{8x^3 - 12x^2 + 18x - 27}{2x - 3 - x^2 + 4x} \), we first need to simplify the denominator.
Start by rearranging the denominator terms: \[ 2x - 3 - x^2 + 4x = -x^2 + 2x + 4x - 3 = -x^2 + 6x - 3. \] Notice that we can factor out a \(-1\): \[ -x^2 + 6x - 3 = -(x^2 - 6x + 3). \]
We now have: \[ \frac{8x^3 - 12x^2 + 18x - 27}{-(x^2 - 6x + 3)}. \]
Next, let's perform the polynomial long division of \(8x^3 - 12x^2 + 18x - 27\) by \(- (x^2 - 6x + 3)\).
- Divide the leading term \(8x^3\) by the leading term of the divisor \(-x^2\): \[ \frac{8x^3}{-x^2} = -8x. \]
- Multiply \(-8x\) by the entire divisor: \[ -8x(-x^2 + 6x - 3) = 8x^3 - 48x^2 + 24x. \]
- Subtract this from the original polynomial: \[ (8x^3 - 12x^2 + 18x - 27) - (8x^3 - 48x^2 + 24x). \] This gives: \[ (-12x^2 + 48x^2) + (18x - 24x) - 27 = 36x^2 - 6x - 27. \]
- Now we need to continue dividing \(36x^2 - 6x - 27\) by \(- (x^2 - 6x + 3)\). Divide: \[ \frac{36x^2}{-x^2} = -36. \]
- Multiply \(-36\) by the entire divisor: \[ -36(-x^2 + 6x - 3) = 36x^2 - 216x + 108. \]
- Subtract this from the remainder: \[ (36x^2 - 6x - 27) - (36x^2 - 216x + 108). \] This gives: \[ (-6x + 216x) + (-27 - 108) = 210x - 135. \]
Putting it all together, we have: \[ -8x - 36 + \frac{210x - 135}{-(x^2 - 6x + 3)}. \] This can be simplified further by factoring or simplifying the remainder term if possible. But overall the expression is simplified to: \[ -8x - 36 - \frac{210x - 135}{x^2 - 6x + 3}. \]
You can factor \(210x - 135\): \[ = 15(14x - 9). \]
Final answer: \[ -8x - 36 - \frac{15(14x - 9)}{-(x^2 - 6x + 3)}. \]
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