Algebra

First, let's multiply by the LCM of the denominators. This will get rid of all the denominators and leave us with a nice clean sheet. We get

\(10\left(x+4\right)=5\left(x-3\right)\left(x+5\right)+2\left(x+7\right)\left(x+5\right)\)

Doing the painful process of multiplying out and combining all like terms, we get

\(10x+40=7x^2+34x-5\)

Moving all terms to one side and combining all like terms, we get

\(7x^2+24x-45=0\)

Now, applying the quadratic equation, we find that

\(x_{1,\:2}=\frac{-24\pm \:\sqrt{24^2-4\cdot \:7\left(-45\right)}}{2\cdot \:7}\)

The 2 values of x are \(x=\frac{3\left(\sqrt{51}-4\right)}{7},\:x=-\frac{3\left(4+\sqrt{51}\right)}{7}\)

Thanks! :)