Can someone explain the steps of this Partial fraction decomposition? 
My thoughts:
$$\frac{(Ax + B)}{(x^2+1)} + \frac{(Cx + D)}{(x^2+4)} = \frac{x}{(x^2+1)(x^2+4)}$$
I combined the left terms
Set the numerator of the combined left term to "x" which is the numerator of the right term
I got
$$4Ax + Cx = x$$
I am not sure what to do next

You have only found the equation for $x$. The full set of equations is derived as follows: 
$$(Ax+B)(x^2+4)+(Cx+D)(x^2+1)\equiv x$$
So compare coefficients of each of the powers of $x$:
$$(A+C)x^3\equiv 0\\(B+D)x^2\equiv 0\\(4A+C)x\equiv x\\(4B+D)1\equiv 0$$
Solving this gives $A=-C,B=-D$. So $$4A-A=1\implies A=\frac13\implies C=-\frac13$$ and $$4B-B=0\implies B=0\implies D=0$$
And this gives the left hand side as 
$$\frac{\frac13 x}{x^2+1}-\frac{\frac 13 x}{x^2+4}$$ as required.
