Question

Perform the operation(s) and simplify completely.

8x3−12x2+18x−272x−3−x2+4x


(1 point)

Answer 1

To simplify the expression \( \frac{8x^3 - 12x^2 + 18x - 27}{2x - 3 - x^2 + 4x} \), we first simplify the denominator.

1. 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.

2. 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:
\[
(2x - 3)(4x^2 + 9)
\]

3. 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:
\[
\frac{(2x - 3)(4x^2 + 9)}{-1(x^2 - 6x + 3)}
\]

4. Notice that \(2x - 3\) cancels out:
\[
= -(4x^2 + 9)/(x^2 - 6x + 3)
\]

Thus, the simplified form of the expression is:
\[
-\frac{4x^2 + 9}{x^2 - 6x + 3}
\]

5. Final answer:
\[
-\frac{4x^2 + 9}{x^2 - 6x + 3}
\]