## To divide the given expression ((x^2 + 2x - 15)/(4x^2)) / ((x^2 - 25)/(2x - 10)), we need to multiply the first term by the reciprocal of the second term.

Step 1: Write the expression as a multiplication problem by flipping the second fraction:

((x^2 + 2x - 15)/(4x^2)) * ((2x - 10)/(x^2 - 25))

Now, let's factor the expressions in the numerators and denominators:

Numerator 1: x^2 + 2x - 15 = (x + 5)(x - 3)

Denominator 1: 4x^2 = 2x * 2x = 2x(2x)

Numerator 2: 2x - 10 = 2(x - 5)

Denominator 2: x^2 - 25 = (x + 5)(x - 5)

Replacing the factors in the expression, we have:

((x + 5)(x - 3)/(2x(2x))) * (2(x - 5)/((x + 5)(x - 5)))

Step 2: Cancel out any common factors:

(x + 5) cancels out in the numerator and denominator.

(x - 5) cancels out in the numerator and denominator.

2 cancels out in the numerator and denominator.

This leaves us with:

((x - 3)/(2x)) * (1/1) = (x - 3)/(2x)

Therefore, the simplified form of ((x^2 + 2x - 15)/(4x^2)) / ((x^2 - 25)/(2x - 10)) is (x - 3)/(2x).