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

1x+2+1x+3+1×2+5x+6
Factor x2+5x+6 using the AC method.
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)
Write each expression with a common denominator of (x+2)(x+3), by multiplying each by an appropriate factor of 1.
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 into one fraction.
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)
Simplify the numerator.
2x+3+2+1(x+2)(x+3)
2x+5+1(x+2)(x+3)
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 of 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)