Sure, let's go through the solution step by step.
We start with the given 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]
First, we'll 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]
Next, let's factorize where possible.
Notice that [tex]\(a^2 - b^2\)[/tex] can be factored as [tex]\((a - b)(a + b)\)[/tex] and [tex]\(5 x^2 + 10 x\)[/tex] can be factored as [tex]\(5 x(x + 2)\)[/tex]:
[tex]\[
\frac{5 x^2}{a-b} \times \frac{(a - b)(a + b)}{5 x (x + 2)} \times \frac{x+2}{2 a b}
\][/tex]
Now, we can cancel out the common factors in the numerators and denominators:
1. [tex]\(a - b\)[/tex] in the second term.
2. [tex]\(5 x\)[/tex] in the first and second terms.
3. [tex]\(x + 2\)[/tex] in the second and third terms.
This gives us:
[tex]\[
\frac{x}{1} \times \frac{a + b}{1} \times \frac{1}{2 a b}
\][/tex]
Simplify by combining the remaining terms:
[tex]\[
\frac{x(a + b)}{2 a b}
\][/tex]
Writing this in a more straightforward format:
[tex]\[
\frac{a x + b x}{2 a b}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\frac{(a x + b x)}{2 a b}
\][/tex]
