Question

Multiply the rational expression and write the product in simplest form: x2+2x−3x2+3x−4⋅x2+6x+8x2−9.(1 point)
Responses

−23
negative Start Fraction 2 over 3 End Fraction

x+2x−3
Start Fraction x plus 2 over x minus 3 End Fraction

−(x+2)x+3
Start Fraction negative left parenthesis x plus 2 right parenthesis over x plus 3 End Fraction

(x+4)(x+2)(x+3)(x−3)

Answer 1

To multiply the given rational expressions and simplify the product, we first need to rewrite the expressions clearly and factor them if possible.

The expression is:

\[
\frac{x^2 + 2x - 3}{x^2 + 3x - 4} \cdot \frac{x^2 + 6x + 8}{x^2 - 9}
\]

### Step 1: Factor each polynomial

1. **Factor \(x^2 + 2x - 3\)**:
\[
x^2 + 2x - 3 = (x + 3)(x - 1)
\]

2. **Factor \(x^2 + 3x - 4\)**:
\[
x^2 + 3x - 4 = (x + 4)(x - 1)
\]

3. **Factor \(x^2 + 6x + 8\)**:
\[
x^2 + 6x + 8 = (x + 4)(x + 2)
\]

4. **Factor \(x^2 - 9\)** (difference of squares):
\[
x^2 - 9 = (x - 3)(x + 3)
\]

### Step 2: Substitute the factored form into the expression

Now we rewrite the expression using these factors:

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

### Step 3: Combine the fractions

Now we can combine the fractions:

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

### Step 4: Cancel common factors

Observe that \( (x + 3) \), \( (x - 1) \), and \( (x + 4) \) appear in both the numerator and the denominator, so we can cancel them:

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

### Step 5: Final form

Thus, the simplest form of the expression is:

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

### Conclusion

So the answer is:

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