Question

Simplify, thank you in advance

Answer 1

So firstly, we will be using the difference of squares with the second fraction's denominator. The difference of squares is
`x^2+y^2=(x+y)(x-y)` . Apply this rule to this expression:
`(3)/(x+4)+(2)/((x+4)(x-4))`

Next, we have to find the LCM, or least common multiple, of both denominators. In this case, the LCM is (x + 4)(x - 4). Multiply the denominators with the quantity that gets the LCM as the denominator, and then multiply by that same amount on their numerators:

`(3)/(x+4)*((x-4))/((x-4))=(3(x-4))/((x+4)(x-4))\\ \\ (2)/((x+4)(x-4))*(1)/(1)=(2)/((x+4)(x-4))\\ \\ (3(x-4))/((x+4)(x-4))+(2)/((x+4)(x-4))`

Now foil 3(x - 4):
`(3x-12)/((x+4)(x-4))+(2)/((x+4)(x-4))`

And lastly, add the numerators up and your final answer will be
`(3x-10)/((x+4)(x-4))`

Answer 2

`\bf \cfrac{3}{x+4}+\cfrac{2}{x^2-16}\implies \cfrac{3}{x+4}+\cfrac{2}{\stackrel{\textit{difference of squares}}{x^2-4^2}}\implies \cfrac{3}{x+4}+\cfrac{2}{(x-4)(x+4)}\\\\\\\stackrel{\textit{so our LCD will be }(x-4)(x+4)}{\cfrac{(x-4)3+(1)2}{(x-4)(x+4)}}\implies \cfrac{3x-12+2}{(x-4)(x+4)}\implies \cfrac{3x-10}{(x-4)(x+4)}`