Answer:
A
Step-by-step explanation:
We are given two functions:
[tex]\displaystyle f(x)=\frac{x-16}{x^2+6x-40}\text{ and } g(x)=\frac{1}{x+10}[/tex]
And we want to find:
[tex]f(x)+g(x)[/tex]
Thus:
[tex]\displaystyle =\frac{x-16}{x^2+6x-40}+\frac{1}{x+10}[/tex]
We can factor the denominator of the first term:
[tex]\displaystyle =\frac{x-16}{(x+10)(x-4)}+\frac{1}{x+10}[/tex]
In order to add the two terms, we must have a common denominator. To achieve this, we can multiply to second term by (x - 4). Therefore:
[tex]\displaystyle =\frac{x-16}{(x+10)(x-4)}+\frac{1}{x+10}\Big(\frac{x-4}{x-4}\Big)[/tex]
Multiply:
[tex]\displaystyle =\frac{x-16}{(x+10)(x-4)}+\frac{x-4}{(x+10)(x-4)}[/tex]
Combine:
[tex]\displaystyle =\frac{(x-16)+(x-4)}{(x+10)(x-4)}[/tex]
Simplify:
[tex]\displaystyle =\frac{2x-20}{(x+10)(x-4)}[/tex]
We can expand the denominator:
[tex]\displaystyle =\frac{2x-20}{x^2+6x-40}[/tex]
Therefore, our answer is A.
