https://file.aiquickdraw.com/mathbot/file/1730314714499pn6g95hp.jpg

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.

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### 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.

What is the sum? \( \frac{7}{(x+1)} + \frac{6}{(x+1)(x-3)} \)

Learn to add rational expressions like \( \frac{7}{(x+1)} + \frac{6}{(x+1)(x-3)} \) by finding the least common denominator and combining terms. This solution covers core algebra concepts, factoring, and simplifying rational expressions. Ideal for students in Grades 8-10.

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