To simplify the expression \( \frac{5x}{x-2} \div \frac{x}{x+2} \), we first rewrite the division as multiplication by the reciprocal:
\[ \frac{5x}{x-2} \div \frac{x}{x+2} = \frac{5x}{x-2} \times \frac{x+2}{x} \]
Next, we can multiply the fractions:
\[ = \frac{5x \cdot (x+2)}{(x-2) \cdot x} \]
Now, simplify the expression. The \( x \) in the numerator and denominator cancels out (provided \( x \neq 0 \)):
\[ = \frac{5(x+2)}{x-2} \]
Now, distribute the \( 5 \):
\[ = \frac{5x + 10}{x-2} \]
Thus, the simplified expression is:
\[ \frac{5x + 10}{x-2} \]
This is the simplest form of the given expression.
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