Denominator help please...?

Question

Preform the indicated operation : Simplify if possible.

1) (x / x^2 + 2x +1) + ( 1 / x^2 + 5x +4)

2) ( x / x^2 +15x +56) - ( 1 / x^2 + 13x + 42)

Answer 1

1) x/(x+1)² + 1/(x+4)(x+1)
[x(x+4) + 1(x+1)]/(x+1)²(x+4)
(x²+5x+1)/(x+1)²(x+4)

2) x/(x+7)(x+8) - 1/(x+6)(x+7)
[x(x+6) - 1(x+8)]/(x+6)(x+7)(x+8)
(x²+5x-8)/(x+6)(x+7)(x+8)

Answer 2

1)

x/x^2 + 2x + 1 + 1/x^2 + 5x + 4

factorise the denominators
first one

x^2 + 2x + 1 = (x +1)^2

second one

x^2 + 5x + 4 =

x^2 + 4x + x + 4 =

x(x + 4) + 1(x + 4) =

(x + 4) (x + 1)
so
x/x^2 + 2x + 1 + 1/x^2 + 5x + 4 =

1/(x + 1)^2 + 1/(x+4)(x +1) =

multiply with (x+4)(x+1)^2/(x+4)(x+1)^2 =

(x + 4 + x + 1)/(x+4)(x+1)^2 =

(2x + 5)/(x+4)(x + 1)^2

2)
x/x^2 + 15x + 56 - 1/x^2 + 13x + 42

x/(x^2 + 8x + 7x + 56) - 1/(x^2 + 7x + 6x + 42) =

x/x(x + 8) + 7(x + 8) - 1/x(x +7) + 6(x +7) =

x/(x + 8)(x + 7) - 1/(x + 7)(x + 6) =

multiply with (x+8)(x+7)(x+6)/ (x+8)(x+7)(x+6)

x(x +6) - (x + 8)/(x+8)(x+7)(x+6) =

(x^2 + 6x - x - 8)/(x+8)(x+7)(x+6) =

(x^2 + 5x -8)/(x+8)(x+7)(x+6)

Answer 3

1) (x / x^2 + 2x +1) + ( 1 / x^2 + 5x +4)
(x + 2x^3 + x^2)/x^2 + (1 + 5x^3 + 4x^2)/x^2
(x + 2x^3 + x^2 + 1 + 5x^3 + 4x^2)/x^2
(7x^3 + 5x^2 + x + 1)/x^2

2) ( x / x^2 +15x +56) - ( 1 / x^2 + 13x + 42)
( x +15x^3 +56x^2)/ x^2 - ( 1+ 13x^3 + 42x^2) / x^2
( x +15x^3 +56x^2 - 1 - 13x^3 - 42x^2) / x^2
( 2x^3 + 14x^2 + x - 1) / x^2