14. $$ \frac{5 x^{2}+10 x}{x+5} \div \frac{x^{2}-4}{x^{2}-25} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{5x^{2} + 10x}{x + 5} \div \frac{x^{2} - 4}{x^{2} - 25}
$$

**Step 1: Rewrite the Division as Multiplication**

Dividing by a fraction is the same as multiplying by its reciprocal. So, the expression becomes:

$$
\frac{5x^{2} + 10x}{x + 5} \times \frac{x^{2} - 25}{x^{2} - 4}
$$

**Step 2: Factor Each Polynomial**

1. **Factor the Numerator $$5x^{2} + 10x$$:**
   $$
   5x^{2} + 10x = 5x(x + 2)
   $$

2. **Factor $$x^{2} - 25$$ (Difference of Squares):**
   $$
   x^{2} - 25 = (x - 5)(x + 5)
   $$

3. **Factor $$x^{2} - 4$$ (Difference of Squares):**
   $$
   x^{2} - 4 = (x - 2)(x + 2)
   $$

**Step 3: Substitute the Factors Back Into the Expression**

$$
\frac{5x(x + 2)}{x + 5} \times \frac{(x - 5)(x + 5)}{(x - 2)(x + 2)}
$$

**Step 4: Cancel Common Factors**

- $$x + 5$$ in the numerator and denominator cancels out.
- $$x + 2$$ in the numerator and denominator cancels out.

This simplifies the expression to:

$$
\frac{5x(x - 5)}{x - 2}
$$

**Final Simplified Form:**

$$
\boxed{\dfrac{5x(x - 5)}{x - 2}}
$$
The simplified form of the expression is $$ \frac{5x(x - 5)}{x - 2} $$.

14. \( \frac{5 x^{2}+10 x}{x+5} \div \frac{x^{2}-4}{x^{2}-25} \)