For the fraction with numerator 8 x to the 7th power plus 10 x to the 6th power plus 20 x to the 5th power plus 33 x to the 4th power plus 23 x cubed plus 37 x squared plus 9 x plus 4 and denominator open paren x plus 1 close paren squared times open paren x squared plus 1 close paren cubed , which expression represents the correct form of the partial fraction decomposition?

The denominator has a repeated linear factor, (x + 1)², so the decomposition needs a constant numerator over each power: A/(x + 1) and B/(x + 1)². It also has the irreducible quadratic factor x² + 1 repeated three times, so each power needs a linear numerator: (Cx + D)/(x² + 1), (Ex + F)/(x² + 1)², and (Gx + H)/(x² + 1)³. This is the form shown in choice D.

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