You can write the given problem as
Substituting x=0, you get A
Substituting x = 1 you get D
When you compare coefficients of same power of x, you get one equation.
In the given problem, there is no tern (x^2 +1) in the denominator, because E and F must be zero.
To find G, equate the coefficient of x^5 on both sides.
TO find H, equate the coefficient of x^4 on both sides.