Question

Express in simplest form: 5x2+10x/x2+7x+12÷x2−4/x2+x−6
.(1 point)
Responses

54
Start Fraction 5 over 4 End Fraction

5x+4
Start Fraction 5 over x plus 4 End Fraction

5xx+4
Start Fraction 5 x over x plus 4 End Fraction

x+45x

Answer 1

To simplify the expression \( \frac{5x^2 + 10x}{x^2 + 7x + 12} \div \frac{x^2 - 4}{x^2 + x - 6} \), we can follow these steps:

1. **Factor the polynomials where possible**.
- \( 5x^2 + 10x = 5x(x + 2) \)
- \( x^2 + 7x + 12 \) factors to \( (x + 3)(x + 4) \)
- \( x^2 - 4 \) is a difference of squares, so it factors to \( (x - 2)(x + 2) \)
- \( x^2 + x - 6 \) factors to \( (x + 3)(x - 2) \)

2. **Rewrite the original expression using these factors**:
\[
\frac{5x(x + 2)}{(x + 3)(x + 4)} \div \frac{(x - 2)(x + 2)}{(x + 3)(x - 2)}
\]

3. **Change the division to multiplication by the reciprocal**:
\[
\frac{5x(x + 2)}{(x + 3)(x + 4)} \times \frac{(x + 3)(x - 2)}{(x - 2)(x + 2)}
\]

4. **Simplify the expression**. Cancel out common factors from the numerator and denominator:
- The \( (x + 2) \) cancels.
- The \( (x + 3) \) cancels.

Now we have:
\[
\frac{5x \cdot (x - 2)}{(x + 4) \cdot (x - 2)}
\]

Again, the \( (x - 2) \) cancels:
\[
\frac{5x}{x + 4}
\]

So, the simplest form of the original expression is:
\[
\frac{5x}{x + 4}
\]

This matches the choice:
**Start Fraction 5 x over x plus 4 End Fraction**, which is the correct answer.