You can put this solution on YOUR website! I assume the expression is:
(In the future, please put parentheses, "(" and ")", around multiple term numerators and denominators. Tutors are more likely to answer if the problem is clear.)
Fractions are easier to add and to simplify if they are factored. So we will start by factoring. The first numerator fits the difference of cubes pattern: . The two denominators trinomials which are very easy to factor. Now our expression is:
Notice the (a-2)'s in the first fraction. We can reduce this fraction and doing so will make the addition and subsequent reducing easier:
Now we proceed with the addition. Of course we need common denominators. With the denominators factors we can see that the first denominator would match the second denominator if it had a factor of (a+2). So we just have to give it this factor by multiplying the denominator and the numerator by (a+2):
To multiply the two numerators we multiply each term of one by each term of the other (and then combine like terms, if any). (I'll leave the deomintors factored for now for reasons that will be clear later.)
Now we can add the two fractions (by adding the two numerators):
Next we would try to factor the numerator to see if the fraction would reduce. Since the only factors that will make the fraction reducible are (a+5) and (a+2), those are the only factors to look for. And the quickest way to check is synthetic division (which I hope you've learned):
-2 | 2 2 2 10 4 8
---- -4 4 -12 4 -16
-----------------------
2 -2 6 -2 8 -8
With a remainder of -8, (a+2) is not a factor.
-5 | 2 2 2 10 4 8
---- -10 40 -210 1000 -5020
--------------------------
2 -8 42 -200 1004 -5012
With a remainder of -5, (a+5) is not a factor.
So even if factors in some way, it will not have factors that will cancel with factors in the denominator. So we we not bother with further attempts to factor. We will just multiply out the denominator:
(