How can I decompose repeated irreducible quadratic factors in partial fractions?

[Stella.T]

I'm a second-year engineering student, and I'm completely stuck on a problem set for my advanced calculus course that focuses on integration techniques. The specific problem requires integrating a rational function using partial fraction decomposition, but the denominator is an irreducible quadratic raised to a power, and my attempts keep leading to a system of equations that seems unsolvable. I've reviewed my algebra, but I'm worried I'm missing a key substitution or a trick to simplify the decomposition before integrating. Can anyone walk through the strategy for handling repeated irreducible quadratic factors?