The technique is called partial fractions:
Now combine the denominators and equate coefficients.
Substitution is often used to solve these types of equations. Note dividing the first equation leaves us with:
Substituting for $-C$ in the third equation gives:
But from the second equation we now have:
Multiply the $4A-2B=1$ equation by $2$ on both sides:
Add both equations above,
Substituting this back into one of the equations that we just used elimination on we get:
I broke it down even more so it would be easier to see the cancellations.