To simplify the expression \( \frac{8x^3 - 12x^2 + 18x - 27}{2x - 3 - x^2 + 4x} \), we first simplify the denominator.
- Combine like terms in the denominator: \[ 2x - 3 - x^2 + 4x = (2x + 4x) - x^2 - 3 = 6x - x^2 - 3 \] This results in: \[ -x^2 + 6x - 3 \]
So the expression can be rewritten as: \[ \frac{8x^3 - 12x^2 + 18x - 27}{-x^2 + 6x - 3} \]
Next, we need to factor both the numerator and the denominator.
- Factor the numerator \( 8x^3 - 12x^2 + 18x - 27 \):
- We can group the terms: \[ (8x^3 - 12x^2) + (18x - 27) \]
- Factor out common factors: \[ 4x^2(2x - 3) + 9(2x - 3) \]
- Now, we factor out \( (2x - 3) \): \[ (2x - 3)(4x^2 + 9) \]
So, our numerator can be written as: \[ (2x - 3)(4x^2 + 9) \]
- Now, rewrite the denominator, which is already factored as: \[ -(x^2 - 6x + 3) \]
Notice, we should actually ensure to factor it correctly.
However, just rewriting it cleanly gives us: \[ -(x^2 - 6x + 3) = -1(x^2 - 6x + 3) \]
Now putting it all together:
The complete expression now looks like: \[ \frac{(2x - 3)(4x^2 + 9)}{-1(x^2 - 6x + 3)} \]
- Notice that \(2x - 3\) cancels out: \[ = -(4x^2 + 9)/(x^2 - 6x + 3) \]
Thus, the simplified form of the expression is: \[ -\frac{4x^2 + 9}{x^2 - 6x + 3} \]
- Final answer: \[ -\frac{4x^2 + 9}{x^2 - 6x + 3} \]
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