To solve the expression [tex]\(\frac{5 x^2}{a-b} \times \frac{a^2-b^2}{5 x^2+10 x} \div \frac{2 a b}{x+2}\)[/tex], we can follow these steps:
1. Understand the components:
- We have three fractions involved in multiplication and division.
2. Rewrite the division as multiplication by the reciprocal:
[tex]\[
\frac{5 x^2}{a-b} \times \frac{a^2-b^2}{5 x^2+10 x} \times \frac{x+2}{2 a b}
\][/tex]
3. Combine the fractions:
Multiply the numerators together and the denominators together:
[tex]\[
\frac{5 x^2 \cdot (a^2 - b^2) \cdot (x + 2)}{(a - b) \cdot (5 x^2 + 10 x) \cdot (2 a b)}
\][/tex]
4. Simplify each part of the expression where possible:
- The term [tex]\(a^2 - b^2\)[/tex] can be factored as [tex]\((a - b)(a + b)\)[/tex].
- The term [tex]\(5 x^2 + 10 x\)[/tex] can be factored as [tex]\(5 x (x + 2)\)[/tex].
So the expression becomes:
[tex]\[
\frac{5 x^2 \cdot (a - b)(a + b) \cdot (x + 2)}{(a - b) \cdot 5 x (x + 2) \cdot 2 a b}
\][/tex]
5. Cancel common terms in the numerator and denominator:
- The [tex]\((a - b)\)[/tex] term cancels out.
- The [tex]\(5 x\)[/tex] cancels with the [tex]\(5 x\)[/tex] in the numerator and denominator.
- The [tex]\((x + 2)\)[/tex] term cancels out.
6. After canceling, we are left with:
[tex]\[
\frac{x \cdot (a + b)}{2 a b}
\][/tex]
Thus, the simplified result is:
[tex]\[
\frac{x(a + b)}{2 a b}
\][/tex]
So, the solution to the expression [tex]\(\frac{5 x^2}{a-b} \times \frac{a^2-b^2}{5 x^2+10 x} \div \frac{2 a b}{x+2}\)[/tex] is:
[tex]\[
\boxed{\frac{x(a + b)}{2 a b}}
\][/tex]
