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The problem involves adding two rational expressions:

\[
\frac{2x + 18}{3x + 7} + \frac{7x + 3}{3x + 7}
\]

### Step 1: Add the Fractions
Since both fractions have the same denominator \( 3x + 7 \), we can directly add the numerators:

\[
\frac{(2x + 18) + (7x + 3)}{3x + 7}
\]

Simplify the numerator:

\[
2x + 18 + 7x + 3 = 9x + 21
\]

So the expression becomes:

\[
\frac{9x + 21}{3x + 7}
\]

### Step 2: Factor the Numerator and Denominator
Factor out common terms in the numerator:

\[
\frac{9(x + 3)}{3x + 7}
\]

The expression cannot be simplified further because \( x + 3 \) and \( 3x + 7 \) are not common factors.

### Step 3: Determine the Excluded Values
To determine which values of \( x \) must be excluded, set the denominator equal to zero and solve for \( x \):

\[
3x + 7 = 0
\]

\[
3x = -7
\]

\[
x = -\frac{7}{3}
\]

Thus, the value \( x = -\frac{7}{3} \) must be excluded because it would make the denominator zero, causing the expression to be undefined.

### Answer:
The correct choice is **A** with \( x \neq -\frac{7}{3} \).

Would you like further details or have any questions?

Here are five related questions to deepen understanding:
1. How do you add fractions with different denominators?
2. What are common methods to simplify rational expressions?
3. Why must certain values of \( x \) be excluded in rational expressions?
4. How would you multiply or divide two rational expressions?
5. What does it mean when a rational expression is undefined?

**Tip:** Always factor both the numerator and denominator to check if any further simplification is possible after combining expressions.

Add the two rational expressions: (2x + 18) / (3x + 7) + (7x + 3) / (3x + 7).

Learn how to add two rational expressions with a common denominator and simplify the result. This step-by-step example includes factoring and identifying excluded values. Suitable for students in Grades 9-12.

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