Question

Perform the operation(s) and simplify completely. 8x3−12x2+18x−272x−3−x2+4x (1 point)

Answer 1

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:
\[
-4x^2 - 9
\]