Question

Express in simplest form:



\[
\frac{5x^2 + 10x}{x^2 + 7x + 12} \div \frac{x^2 - 4}{x^2 + x - 6}
\]



A) \(\frac{5}{x + 4}\)

B) \(\frac{5}{4}\)

C) \(\frac{5x}{x + 4}\)

D) \(\frac{x + 4}{5x}\)

Answer 1

To solve the expression

\[
\frac{5x^2 + 10x}{x^2 + 7x + 12} \div \frac{x^2 - 4}{x^2 + x - 6},
\]

we will first rewrite the division as a multiplication by the reciprocal:

\[
\frac{5x^2 + 10x}{x^2 + 7x + 12} \cdot \frac{x^2 + x - 6}{x^2 - 4}.
\]

### Step 1: Factor each polynomial.

**Numerator: \(5x^2 + 10x\)**
\[
5x^2 + 10x = 5x(x + 2).
\]

**Denominator: \(x^2 + 7x + 12\)**
To factor \(x^2 + 7x + 12\), we need two numbers that multiply to \(12\) and add to \(7\). These numbers are \(3\) and \(4\):
\[
x^2 + 7x + 12 = (x + 3)(x + 4).
\]

**Numerator: \(x^2 + x - 6\)**
To factor \(x^2 + x - 6\), we need two numbers that multiply to \(-6\) and add to \(1\). These numbers are \(3\) and \(-2\):
\[
x^2 + x - 6 = (x - 2)(x + 3).
\]

**Denominator: \(x^2 - 4\)**
This is a difference of squares:
\[
x^2 - 4 = (x - 2)(x + 2).
\]

### Step 2: Rewrite the expression in factored form.

Substituting the factored forms back into the expression gives us:

\[
\frac{5x(x + 2)}{(x + 3)(x + 4)} \cdot \frac{(x - 2)(x + 3)}{(x - 2)(x + 2)}.
\]

### Step 3: Simplify the expression.

Now we can cancel out any common factors:
- \(x + 2\) cancels with \(x + 2\) from the denominator.
- \(x - 2\) cancels with \(x - 2\) from the numerator.

This leaves us with:

\[
\frac{5x}{(x + 4)} \cdot 1.
\]

### Conclusion

Thus we have simplified the entire expression to:

\[
\frac{5x}{x + 4}.
\]

The answer is

\(\boxed{\frac{5x}{x + 4}}\).

Thus the correct option is C) \(\frac{5x}{x + 4}\).