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! :)