In order to combine fractions the denominators must be equal. #x^2-9# and #x-3# aren't, so let's see what we can do about that. First we need to see what it would take to make them equal. #x^2-9# can be expanded to #(x+3)(x-3)#. Both denominators should have these components, and #x-3# is missing #x+3#. We need to multiply #x/(x-3)# on the top and bottom by #x+3#. #x/(x-3)# #*(x+3)/(x+3)# becomes #(x^2+3x)/((x-3)(x+3))#, or #(x^2+3x)/(x^2-9)#. Now we can combine #(2x)/(x^2-9)# with #(x^2+3x)/(x^2-9)#.
When we add or subtract fractions, the first step is to have the denominators equal. Then we just add or subtract the numerators straight accross and leave the denominators alone. The final answer should have the same denominator as on the other side of the equal sign.
#(2x)/(x^2-9) + (x^2+3x)/(x^2-9) = (x^2+5x)/(x^2-9)#. This cannot be simplified further, so we'll leave it here. Nice work.
