To solve for the expression [tex]\( f(x) + g(x) \)[/tex], let's follow these steps:
We start with the given functions:
[tex]\[ f(x) = \frac{x-16}{x^2 + 6x - 40} \][/tex]
[tex]\[ g(x) = \frac{1}{x+10} \][/tex]
### Step 1: Find a Common Denominator
To add these two fractions, [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], we need a common denominator. The denominators are:
[tex]\[ x^2 + 6x - 40 \][/tex] and [tex]\[ x + 10 \][/tex]
The common denominator will be the product of these two expressions:
[tex]\[ (x^2 + 6x - 40)(x + 10) \][/tex]
### Step 2: Rewrite Each Fraction with the Common Denominator
Next, we rewrite both fractions with this common denominator.
#### For [tex]\( f(x) \)[/tex]:
The numerator of [tex]\( f(x) \)[/tex] needs to be adjusted by multiplying it by [tex]\( x + 10 \)[/tex]:
[tex]\[ f(x) = \frac{(x - 16)(x + 10)}{(x^2 + 6x - 40)(x + 10)} \][/tex]
#### For [tex]\( g(x) \)[/tex]:
The numerator of [tex]\( g(x) \)[/tex] needs to be adjusted by multiplying it by [tex]\( x^2 + 6x - 40 \)[/tex]:
[tex]\[ g(x) = \frac{1 \cdot (x^2 + 6x - 40)}{(x + 10)(x^2 + 6x - 40)} \][/tex]
### Step 3: Add the Fractions
With the common denominator established, add the fractions by summing their numerators:
[tex]\[ \frac{(x - 16)(x + 10) + (x^2 + 6x - 40)}{(x^2 + 6x - 40)(x + 10)} \][/tex]
### Step 4: Simplify the Numerator
First, expand [tex]\((x - 16)(x + 10)\)[/tex]:
[tex]\[ (x - 16)(x + 10) = x^2 + 10x - 16x - 160 = x^2 - 6x - 160 \][/tex]
Now, add this result to [tex]\( (x^2 + 6x - 40) \)[/tex]:
[tex]\[ x^2 - 6x - 160 + x^2 + 6x - 40 \][/tex]
Combine like terms:
[tex]\[ = 2x^2 - 200 \][/tex]
### Step 5: Write the Sum
We now have:
[tex]\[ f(x) + g(x) = \frac{2x^2 - 200}{(x^2 + 6x - 40)(x + 10)} \][/tex]
### Step 6: Simplify the Result
To match the form of the given options, factor out a common term in the numerator:
[tex]\[ 2x^2 - 200 = 2(x^2 - 100) = 2(x - 10)(x + 10) \][/tex]
This can simplify to:
[tex]\[ \frac{2(x - 10)(x + 10)}{(x^2 + 6x - 40)(x + 10)} \][/tex]
Cancel out the common factor [tex]\((x + 10)\)[/tex] in the numerator and denominator:
[tex]\[ \frac{2(x - 10)}{x^2 + 6x - 40} \][/tex]
Thus, the final simplified form of [tex]\( f(x) + g(x) \)[/tex] is:
[tex]\[ \frac{2x - 20}{x^2 + 6x - 40} \][/tex]
### Step 7: Compare with the Given Options
From the obtained simplified expression:
[tex]\[ \frac{2x - 20}{x^2 + 6x - 40} \][/tex]
We see that it matches Option C:
[tex]\[ \frac{2x - 20}{x^2 + 6x - 40} \][/tex]
Therefore, the correct expression equal to [tex]\( f(x) + g(x) \)[/tex] is:
[tex]\[ \boxed{C} \][/tex]
