To simplify the given expression:
[tex]\[
\frac{x^2 - 2x}{x^2 - 10x + 25} \div \frac{6x^2 - 12x}{x^2 - 25}
\][/tex]
we follow these steps:
1. Simplify each part of the given fractions:
- The numerator of the first fraction:
[tex]\[
x^2 - 2x = x(x - 2)
\][/tex]
- The denominator of the first fraction:
[tex]\[
x^2 - 10x + 25 = (x - 5)^2
\][/tex]
- The numerator of the second fraction:
[tex]\[
6x^2 - 12x = 6x(x - 2)
\][/tex]
- The denominator of the second fraction:
[tex]\[
x^2 - 25 = (x - 5)(x + 5)
\][/tex]
2. Rewrite the expression with the simplified parts:
[tex]\[
\frac{x(x - 2)}{(x - 5)^2} \div \frac{6x(x - 2)}{(x - 5)(x + 5)}
\][/tex]
3. Convert the division of fractions into a multiplication by inverting the second fraction:
[tex]\[
\frac{x(x - 2)}{(x - 5)^2} \times \frac{(x - 5)(x + 5)}{6x(x - 2)}
\][/tex]
4. Simplify by canceling out common factors:
- The [tex]\(x(x - 2)\)[/tex] terms cancel out in the numerator and denominator.
- The [tex]\((x - 5)\)[/tex] term cancels out with one of the [tex]\((x - 5)\)[/tex] terms in the denominator.
Thus, we are left with:
[tex]\[
\frac{x + 5}{6(x - 5)}
\][/tex]
Therefore, the simplest form of the given expression is:
[tex]\[
\boxed{\frac{x+5}{6(x-5)}}
\][/tex]
