To solve this expression, we need to add the two fractions:

$\frac{7}{(x+1)} + \frac{6}{(x+1)(x-3)}$

Step 1: Identify the Least Common Denominator (LCD)

The LCD for these fractions is $(x+1)(x-3)$.

Step 2: Rewrite Each Fraction with the LCD as Denominator

1. The first fraction, $\frac{7}{(x+1)}$, can be rewritten to have the LCD as its denominator by multiplying both the numerator and denominator by $(x - 3)$: $\frac{7(x - 3)}{(x+1)(x-3)} = \frac{7x - 21}{(x+1)(x-3)}$

2. The second fraction, $\frac{6}{(x+1)(x-3)}$, already has the LCD as its denominator, so we don’t need to change it: $\frac{6}{(x+1)(x-3)}$

Step 3: Add the Numerators

Now that both fractions have the same denominator, we can add their numerators: $\frac{7x - 21 + 6}{(x+1)(x-3)} = \frac{7x - 15}{(x+1)(x-3)}$

Final Answer

The sum is: $\frac{7x - 15}{x^2 - 2x - 3}$ since $(x+1)(x-3) = x^2 - 2x - 3$.

Let me know if you have any questions or if you'd like further details.

Additional Practice Questions

1. How would you find the sum if the denominators were different?
2. What is the process for simplifying the expression $\frac{14}{x^2 - 2x - 3} + \frac{12}{x^2 - 2x - 3}$?
3. How can you identify the LCD for expressions with multiple terms in the denominator?
4. What are the steps for factoring quadratic expressions like $x^2 - 2x - 3$?
5. How would the process change if the fractions involved subtraction instead of addition?

Tip

When adding fractions, always check if the terms in the numerator can be simplified after combining them; sometimes, common factors can be factored out to further simplify the result.