Please fill in the first box and also second box if option i

Please fill in the first box, and also second box (if option is A)

Use the division algorithm to rewrite the improper fraction as the sum of a quotient and a proper fraction. Find the partial fraction decomposition of the proper fraction. Finally, express the improper fraction as the sum of a quotient and the partial fraction decomposition. x^4 - 5x^2 + x - 4/x^2 + 4x + 4 Using the division algorithm, write the improper fraction as the sum of the quotient and a proper fraction. x^4 - 5x^2 +x -4/x^2 + 4x + 4 (Simplify your answer. Use integers or fractions for any terms in the expression.) Using partial fraction decomposition, write the above expression as the sum of a quotient and a partial fraction decomposition. If the expression cannot be decomposed further, say so. Select the correct option below and fill in the answer box if needed. After partial fraction decomposition, x^4 - 5x^2 +x -4/x^2 + 4x + 4 is equivalent to (Simplify your answer. Use integers or fractions for any terms in the expression.) The expression found above cannot be decomposed further.

Solution

x^4 - 5x^2 +x - 4 can be written as (x^ 4 - 5x^2 +12 x + 28) - ( 11 x + 32) but (x^ 4 - 5x^2 +12 x + 28) can be factorised with the denominator x^2 + 4x + 4 and written as ( x^2 + 4x+ 4) * ( x^2 - 4x + 7)

Thus the fraction x^4 - 5x^2 +x - 4 / ( x^2 + 4x + 4) becomes =

( x^2 + 4x+ 4) * ( x^2 - 4x + 7) / ( x^2 + 4x+ 4) - (11 x + 32) / ( x^2 + 4x+ 4) which is =
( x^2 - 4x + 7) - [ (11 x + 32) / ( x^2 + 4x+ 4) ]
Thus the given fraction can be reduced to ( x^2 - 4x + 7) - [ (11 x + 32) / ( x^2 + 4x+ 4) ]