Answer
$$x=5+\sqrt{53},\:x=5-\sqrt{53}$$
Work Step by Step
Solving by plugging in the given value and then using the quadratic formula, we find:
$$ 1\cdot \:x\left(x+4\right)=\frac{7}{x}x\left(x+4\right)+\frac{7}{x+4}x\left(x+4\right)\\ x\left(x+4\right)=7\left(x+4\right)+7x\\ x^2-10x-28=0\\ x_{1,\:2}=\frac{-\left(-10\right)\pm \sqrt{\left(-10\right)^2-4\cdot \:1\left(-28\right)}}{2\cdot \:1}\\ x=5+\sqrt{53},\:x=5-\sqrt{53}$$