You're on the right track, but don't forget that you're working with an equation. If you had been dealing only with an expression, then you'd be stuck with working with common denominators. But you're not! This means that you can multiply through to clear the denominators:
a(x−b)(x+c)+b(x+a)(x+c)=(a+b)(x+a)(x−b)
It's still painful, but not quite so much.
a(x2−bx+cx−bc)+b(x2+ax+cx+ac)=(a+b)(x2+ax−bc−ab)
ax2−abc+acx−abc+bx2+abc+bcx+abc=ax2+bx2+a2x−abc−a2b+abx−b2c−ab2
Simplify where possible. Then gather everything onto one side of the equation, with zero on the other. You'll see that all this is, is a painful quadratic equation. And the Quadratic Formula can always fix that for you.