1/x^2+3x+2 + 2/x^2-1

I first did 1/x^2+2x+x+2 +2/(x-1)(x+1)
1/x(x+2)x+2 +2/(x-1)(x+1)
1/x+2(x+1) +2/(x-1)(x+1)
x-1+2(x+2)/(x+2)(x-1)(x+1)
x-1+2x+4/(x+2)(x-1)(x+1)
3x+3/(x+2)(x-1)(x+1)
3(x+1)/(x+2)(x-1)(x+1) the x+1 cancel out so
3/(x+2)(x-1)
3/x^2-x+2x-2
answer 3/x^2+x-2
Is this correct? thank you for checking my work

When typing questions like this on this webpage, you MUST use brackets

to determine the correct order of operation.
I will interpret your 1/x^2+3x+2 + 2/x^2-1 as
1/(x^2+3x+2) + 2/(x^2-1)
not even looking at what you did, you can find your errors later
= 1/( (x+2)(x+1) ) + 2/( (x-1)(x+1) )
so the LCD is (x-1)(x+1)(x+2)

= (x-1)/( (x-1)(x+1)(x+2) ) + 2(x+2)/( (x-1)(x+1)(x+2) )
= (3x + 3)/( (x-1)(x+1)(x+2) )
= 3(x+1)/( (x-1)(x+1)(x+2) )
= 3/( (x-1)(x+2) ) , x ≠ -1

Your answer is correct, because you wrote it out
on paper in proper fraction form. Unfortunately we
cannot type fractions here, so as I said before, brackets are needed.

Thank you

To confirm the correctness of your work on the expression 1/x^2+3x+2 + 2/x^2-1, let's go through the steps to simplify it:

1. Start with the given expression: 1/x^2+3x+2 + 2/x^2-1
2. Factor the denominators: 1/x^2+2x+x+2 +2/(x-1)(x+1)
3. Simplify the fractions: 1/x(x+2)+x+2 +2/(x-1)(x+1)
4. Combine the fractions over the common denominator: x-1+2(x+2)/(x+2)(x-1)(x+1)
5. Simplify the resulting expression: x-1+2x+4/(x+2)(x-1)(x+1)
6. Combine like terms: 3x+3/(x+2)(x-1)(x+1)
7. Further simplify if possible: 3(x+1)/(x+2)(x-1)(x+1)

At this point, you correctly canceled out the (x+1) terms in the numerator and denominator.

8. Final answer: 3/(x+2)(x-1)

So, your final answer of 3/(x+2)(x-1) is indeed correct. Well done!