given that 4x^2-6x+9=A(x-1)(2x+1)+B(x-1)+C for all values of x, find the values of A, B and C
just expand the right side and the equate coefficients of like powers. If the polynomials are identical, all the powers of x must have the same coefficients.
You can see right off that the right side will have 2Ax^2+...
So, 4 = 2A and A=2
and so on.