Find the partial fraction decomposition of 4x²+5x-9/(x³- 6x-9)

Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions. This method is particularly useful for integrating rational functions or simplifying complex algebraic expressions. The goal is to break down a fraction into components that are easier to work with, typically involving linear or irreducible quadratic factors in the denominator.

Polynomial long division is a process used to divide one polynomial by another, similar to numerical long division. When the degree of the numerator is greater than or equal to the degree of the denominator, this step is necessary to simplify the expression before applying partial fraction decomposition. The result of this division can be expressed as a polynomial plus a remainder, which can then be decomposed.

Factoring polynomials involves expressing a polynomial as a product of its simpler polynomial factors. This is crucial in partial fraction decomposition, as the denominators must be factored into linear or irreducible quadratic factors to set up the decomposition correctly. Understanding how to factor polynomials allows for the identification of the appropriate form for the partial fractions.