Let's start by simplifying the expression step by step:
Given expression: \[ \frac{8x^3 - 12x^2 + 18x - 27}{2x - 3 - x^2 + 4x} \]
First, we can simplify the denominator: \[ 2x - 3 - x^2 + 4x = -x^2 + 6x - 3 \]
Now, we rewrite the entire expression as: \[ \frac{8x^3 - 12x^2 + 18x - 27}{-x^2 + 6x - 3} \]
Next, we can factor both the numerator and the denominator if possible.
Step 1: Factor the numerator
The numerator is: \[ 8x^3 - 12x^2 + 18x - 27 \]
We can group the terms: \[ (8x^3 - 12x^2) + (18x - 27) \] Factoring out common terms: \[ 4x^2(2x - 3) + 9(2x - 3) \]
Now factor out the common factor \( (2x - 3) \): \[ (2x - 3)(4x^2 + 9) \]
Step 2: Factor the denominator
The denominator is: \[ -x^2 + 6x - 3 \] To factor it, we can look for roots using the quadratic formula, or observe if it can be factored directly.
The quadratic \( -x^2 + 6x - 3 \) can be rewritten as: \[ -(x^2 - 6x + 3) \]
Now, we can evaluate if the quadratic can be factored directly. Since \(-x^2 + 6x - 3\) doesn't factor nicely, we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Where \(a = -1\), \(b = 6\), and \(c = -3\): \[ x = \frac{-6 \pm \sqrt{6^2 - 4(-1)(-3)}}{2(-1)} = \frac{-6 \pm \sqrt{36 - 12}}{-2} = \frac{-6 \pm \sqrt{24}}{-2} = \frac{-6 \pm 2\sqrt{6}}{-2} = 3 \mp \sqrt{6} \]
So we see the roots of the denominator, which doesn’t yield an obvious factorization with integers.
Step 3: Final Simplification
Putting it all together: \[ \frac{(2x - 3)(4x^2 + 9)}{-(x^2 - 6x + 3)} \]
Since \(2x - 3\) is a common factor between the numerator and the negative version of the denominator, we can simplify: \[ \frac{(2x - 3)(4x^2 + 9)}{-(2x - 3)(-x^2 + 6x - 3)} = - (4x^2 + 9) \]
Thus, the final simplified expression is: \[ -(4x^2 + 9) \]
So the answer is: \[ -4x^2 - 9 \]
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