How does one go about decomposing this fraction?
First, take the common factor in the denominator out front. Then factor the remaining quadratic.
Then do the set-up for partial-fraction decomposition: Create fractions with the linear factors (with appropriate powers) as the denominators and letters (A, B, C, etc) as numerators. Sum these fractions (but do not combine them!), and set this sum equal to the original fraction.
Then multiply through by the denominator, and solve the resulting equation for the values of the letters. (This is often done by "equating coefficients".)
If you get stuck, please reply showing your steps and reasoning so far, starting with your factorization of the denominator. Thank you!