a. Factorize the denominator:
Then we're looking for such that
If , then ; if , then . So we have
as required.
b. Same setup as in (a):
We want to find such that
Quick aside: for the second term, since the denominator has degree 2, we should be looking for another constant such that the numerator of the second term is . We always want the polynomial in the numerator to have degree 1 less than the degree of the denominator. But we would end up determining anyway.
If , then ; if , then . Expanding everything on the right then gives
which tells us and ; in both cases, we get . Then
as required.