To find the sum of the rational expressions [tex]\(\frac{x}{x+1} + \frac{6x}{x+5}\)[/tex], we need to follow these steps:
1. Find the common denominator:
Identify the denominators of the two rational expressions:
[tex]\[
x + 1 \quad \text{and} \quad x + 5
\][/tex]
The common denominator is the product of these two denominators:
[tex]\[
(x + 1)(x + 5)
\][/tex]
2. Rewrite each fraction with the common denominator:
For [tex]\(\frac{x}{x+1}\)[/tex], multiply both the numerator and denominator by [tex]\((x + 5)\)[/tex]:
[tex]\[
\frac{x \cdot (x + 5)}{(x+1)(x+5)}
\][/tex]
This gives:
[tex]\[
\frac{x(x + 5)}{(x+1)(x+5)} = \frac{x^2 + 5x}{(x+1)(x+5)}
\][/tex]
For [tex]\(\frac{6x}{x+5}\)[/tex], multiply both the numerator and denominator by [tex]\((x + 1)\)[/tex]:
[tex]\[
\frac{6x \cdot (x + 1)}{(x+5)(x+1)}
\][/tex]
This gives:
[tex]\[
\frac{6x(x + 1)}{(x+5)(x+1)} = \frac{6x^2 + 6x}{(x+5)(x+1)}
\][/tex]
3. Add the fractions:
Combine the fractions by adding their numerators over the common denominator:
[tex]\[
\frac{x^2 + 5x}{(x+1)(x+5)} + \frac{6x^2 + 6x}{(x+1)(x+5)} = \frac{x^2 + 5x + 6x^2 + 6x}{(x+1)(x+5)}
\][/tex]
4. Combine like terms in the numerator:
[tex]\[
x^2 + 5x + 6x^2 + 6x = 7x^2 + 11x
\][/tex]
5. Write the final sum:
The sum of the rational expressions is:
[tex]\[
\frac{7x^2 + 11x}{(x+1)(x+5)}
\][/tex]
The common denominator can be expanded:
[tex]\[
(x+1)(x+5) = x^2 + 6x + 5
\][/tex]
Therefore, the final sum is:
[tex]\[
\frac{7x^2 + 11x}{x^2 + 6x + 5}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{\frac{7x^2 + 11x}{x^2 + 6x + 5}} \quad \text{which corresponds to option} \quad \text{B}
\][/tex]
