Question

Multiply the rational expression and write the product in simplest form: x2+2x−3x2+3x−4⋅x2+6x+8x2−9 .(1 point) Responses (x+4)(x+2)(x+3)(x−3) Start Fraction left parenthesis x plus 4 right parenthesis left parenthesis x plus 2 right parenthesis over left parenthesis x plus 3 right parenthesis left parenthesis x minus 3 right parenthesis End Fraction −(x+2)x+3 Start Fraction negative left parenthesis x plus 2 right parenthesis over x plus 3 End Fraction −23 negative Start Fraction 2 over 3 End Fraction x+2x−3

Answer 1

To multiply the rational expressions \( \frac{x^2+2x-3}{x^2+3x-4} \cdot \frac{x^2+6x+8}{x^2-9} \), let's first factor each of the expressions.

1. **Factor \(x^2 + 2x - 3\)**:
- This factors to \((x + 3)(x - 1)\).

2. **Factor \(x^2 + 3x - 4\)**:
- This factors to \((x + 4)(x - 1)\).

3. **Factor \(x^2 + 6x + 8\)**:
- This factors to \((x + 2)(x + 4)\).

4. **Factor \(x^2 - 9\)**:
- This factors to \((x - 3)(x + 3)\).

Now, substituting these factors into the expression, we have:

\[
\frac{(x + 3)(x - 1)}{(x + 4)(x - 1)} \cdot \frac{(x + 2)(x + 4)}{(x - 3)(x + 3)}
\]

Next, we can simplify the expression by canceling out common factors in the numerator and denominator:

- The \((x - 1)\) terms cancel.
- The \((x + 4)\) terms cancel.
- The \((x + 3)\) terms also cancel.

This leaves us with:

\[
\frac{(x + 2)}{(x - 3)}
\]

Now, there are no more common factors, and the expression \(\frac{(x + 2)}{(x - 3)}\) is already in simplified form.

Thus, the final answer is:

\[
\frac{x + 2}{x - 3}
\]

The correct response from the given options would be equivalent to \(-\frac{(x + 2)}{(x + 3)}\) is:

\(-\frac{2}{3} \cdot \frac{x + 2}{x - 3}\)