2(x+5)/(x+5)(x-2) = 3(x-2)/(x-2)(x+5) + 10/(x+5)(x-2)

I know I have to get

2(x+5) = 3(x-2) but I don't know the intermediate steps. I tried canceling out to get 2/(x-2) = 3 + 10 / (x+5) and gross multiplying but that wouldn't make sense.

To simplify the given equation, we need to eliminate the denominators. Let's go through the intermediate steps together.

Starting with the given equation:
2(x+5)/(x+5)(x-2) = 3(x-2)/(x-2)(x+5) + 10/(x+5)(x-2)

We notice that both the denominators (x+5)(x-2) are common to all terms, so we can eliminate them by multiplying both sides of the equation by (x+5)(x-2):

[(x+5)(x-2)] * [2(x+5)/(x+5)(x-2)] = [(x+5)(x-2)] * [3(x-2)/(x-2)(x+5) + 10/(x+5)(x-2)]

Now, let's simplify this equation step-by-step.

On the left side, the (x+5)(x-2) cancels out with the denominator on the right side. We are left with:
2(x+5) = 3(x-2) + 10/(x+5)

Next, let's distribute the terms:
2x + 10 = 3x - 6 + 10/(x+5)

Now, let's move all the terms to one side by subtracting 2x and adding 6 from both sides:
10 - 6 = 3x - 2x + 10/(x+5)

Simplifying, we have:
4 = x + 10/(x+5)

Finally, if we multiply both sides by (x+5) to eliminate the fraction, we get:
4(x+5) = x(x+5) + 10

Expanding both sides:
4x + 20 = x^2 + 5x + 10

Now, let's move all the terms to one side and rewrite it in standard quadratic form:
x^2 + 5x - 4x - 10 - 20 = 0

Simplifying:
x^2 + x - 30 = 0

Now, you can solve this quadratic equation by factoring, completing the square, or using the quadratic formula.

I hope this explanation helps you understand the intermediate steps required to simplify the given equation.