This part is called composing. We combine the fractions into a larger fraction. We then combine like terms.
This part is called decomposition. We are given a large fraction and are trying to find the small fractions that multiplied to make it.
Simply type the left part into the calculator and using rref you will get the answer on the right, which is your A, B, and C.
This is what you started with when you plug A, B, and C, back together. CONGRATS.