Simplify [1÷(x2+3x+2)]+[1÷(x2+5x+6)]

Simplify $[1÷(x^2+3x+2\left)\right]+[1÷(x^2+5x+6\left)\right]$

Answer Details

$[1÷(x^2+3x+2\left)\right]+[1÷(x^2+5x+6\left)\right]$
= $1÷(x^2+3x+2)+[1÷(x^2+5x+6\left)\right]$
= $[1÷((x^2+x)+(2x+2)\left)\right]+[1÷((x^2+3x)+(2x+6)\left)\right]$
= [1 ÷ (x(x + 2) + 2(x +1))] + [1 ÷ (x(x + 3) +2(x + 3) )]
= [1 ÷ (x + 1)(x + 2)] + [1 ÷ ((x + 3) + (x + 2))]
=((x + 3) + (x + 1)) ÷ (x + 1)(x + 2)(x + 3)
Using the L.C.M
=((x + x + 3 + 1)) ÷ (x + 1)(x + 2)(x + 3)
=(2x+4)/(x+1)(x+2)(x+3) =2(x+2)/(x+1)(x+2)(x+3)
= $\frac{2}{(x+1)(x+3)}$