1x+2+1x+3+1×2+5x+6

Consider the form x2+bx+c. Find a pair of integers whose product is c and whose sum is b. In this case, whose product is 6 and whose sum is 5.

2,3

Write the factored form using these integers.

1x+2+1x+3+1(x+2)(x+3)

1x+2+1x+3+1(x+2)(x+3)

To write 1x+2 as a fraction with a common denominator, multiply by x+3x+3.

1x+2⋅x+3x+3+1x+3+1(x+2)(x+3)

To write 1x+3 as a fraction with a common denominator, multiply by x+2x+2.

1x+2⋅x+3x+3+1x+3⋅x+2x+2+1(x+2)(x+3)

Multiply 1x+2 and x+3x+3.

x+3(x+2)(x+3)+1x+3⋅x+2x+2+1(x+2)(x+3)

Multiply 1x+3 and x+2x+2.

x+3(x+2)(x+3)+x+2(x+3)(x+2)+1(x+2)(x+3)

Reorder the factors of (x+3)(x+2).

x+3(x+2)(x+3)+x+2(x+2)(x+3)+1(x+2)(x+3)

x+3(x+2)(x+3)+x+2(x+2)(x+3)+1(x+2)(x+3)

Combine the numerators over the common denominator.

x+3+x+2(x+2)(x+3)+1(x+2)(x+3)

Combine the numerators over the common denominator.

x+3+x+2+1(x+2)(x+3)

x+3+x+2+1(x+2)(x+3)

Add x and x.

2x+3+2+1(x+2)(x+3)

Add 3 and 2.

2x+5+1(x+2)(x+3)

Add 5 and 1.

2x+6(x+2)(x+3)

Factor 2 out of 2x+6.

Factor 2 out of 2x.

2(x)+6(x+2)(x+3)

Factor 2 out of 6.

2x+2⋅3(x+2)(x+3)

Factor 2 out of 2x+2⋅3.

2(x+3)(x+2)(x+3)

2(x+3)(x+2)(x+3)

2(x+3)(x+2)(x+3)

Cancel the common factor.

2(x+3)(x+2)(x+3)

Rewrite the expression.

2x+2

2x+2

Simplify 1/(x+2)+1/(x+3)+1/(x^2+5x+6)