How do I find the partial fraction decomposition of 2x 7x

Solution

x² + 2x +1 ) 2x² + 7x + 9 ( 2
.................. 2x² + 4x + 2
                    ---------------      
.................. .........3x + 7       

2x² + 7x + 9 = 2(x² + 2x +1) + (3x +7)

==> (2x² + 7x + 9)/(x² + 2x +1) = [ 2(x² + 2x +1) + (3x +7)]/(x² + 2x +1)

==> 2(x² + 2x +1)/(x² + 2x +1)     + (3x +7)/(x² + 2x +1)

==> 2 +   (3x +7)/(x² + 2x +1)

consider (3x +7)/(x² + 2x +1) = (3x +7)/(x +1)²

let (3x +7)/(x +1)² = A/(x +1) + B/(x +1)²

==> 3x +7 = A(x +1) + B

==> 3x +7 = Ax + A + B

Comparing coefficients of like terms

==> A = 3 , A +B = 7

==> B = 7 - A = 7 - 3 = 4

==> (3x +7)/(x +1)² = 3/(x +1) + 4/(x +1)²

==> 2x² + 7x + 9)/(x² + 2x +1) = 2 + 3/(x +1) + 4/(x +1)²