Find numbers A & B such that for all x, x + 2 = A(x - 3) + B(x + 1)
x +2 = A(x -3) +B(x +1) ------(i)
x +2 = Ax -3A +Bx +B
x +2 = (A +B)x +(-3A +B) ------(ii)
For (ii) to be true,
a)the coefficient of x in the LHS must be equal to the coefficient of x in the RHS.
Or, 1 = A +B -----------(iii)
b) and the constants on both sides of the equation (ii) must also be equal.
Or,
2 = -3A +B ------------(iv)
Now solve (iii) and (iv) simultaneously to find A and B.
You'd get
A = -1/4
B = 5/4