Final answer:
Partial fraction decomposition of the expression involves expressing the rational function as a sum of simpler fractions, each with denominators that are factors of the original denominator. The decomposition will include terms with (x + 1) and (x^2 + 1), each raised to their respective powers with unknown constants which are determined by equating coefficients or comparing sums.
Step-by-step explanation:
To perform a partial fraction decomposition of the given rational expression 8x7 + 10x6 + 20x5 + 33x4 + 23x3 + 37x2 + 9x + 4 over (x + 1)2(x2 + 1)3, we first need to recognize the nature and multiplicity of the polynomial factors in the denominator. Here, we have (x + 1) to the power of 2 and (x2 + 1) to the power of 3. Using the partial fraction decomposition method, we will express our original fraction as a sum of simpler fractions where each term's denominator has only factors of the original denominator to the first power.
The general form of the decomposition considering the multiplicity of the factors will be:
A/(x + 1) + B/(x + 1)2 + C/(x2 + 1) + D/(x2 + 1)2 + E/(x2 + 1)3
Each fraction is expanded until it properly represents all possible partial fractions that can occur from the original fraction. The constants A, B, C, D, and E are then found by multiplying both sides by the common denominator to clear the fractions and equating the coefficients of corresponding powers of x, or by another method, such as comparing the sum of the fractions to the original fraction.
To find the value of each constant, one would typically substitute various values of x into the equation and solve the resulting system of equations. This process will yield the specific numerator values for each fraction in the decomposition.