Question

What is the sum? (1/x+2)+(1/x+3)+(1/x^2+5x+6)

a. 2/x+2
b. 1/x+1
c. -2/(x+3)(x+2)
d. -2(x-3)/(x+3)(x+2)

Answer 1

A.) 2/x+2 just took the test and got %100

Answer 2

Factor the denominator of the third fraction:

`x^2+5x+6=x^2+3x+2x+6=x(x+3)+2(x+3)=(x+3)(x+2).`

Then

`(1)/(x+2)+(1)/(x+3)+(1)/(x^2+5x+6)=(1)/(x+2)+(1)/(x+3)+(1)/((x+2)(x+3)).`

The common denominator is
`(x+2)(x+3),` so

`(1)/(x+2)+(1)/(x+3)+(1)/((x+2)(x+3))=(1\cdot (x+3)+1\cdot (x+2)+1)/((x+2)(x+3))=(2x+6)/((x+2)(x+3))=\\ \\(2(x+3))/((x+2)(x+3))=(2)/(x+2).`

Answer: correct choice is A.