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To solve the given problem, we need to add the rational expressions:

\[
\frac{x + 2}{x^2 + 3x - 28} + \frac{x + 3}{x^2 - 8x + 16}.
\]

### Step 1: Factor the Denominators
1. **Factor \( x^2 + 3x - 28 \)**:
   - This expression factors into \( (x + 7)(x - 4) \).

2. **Factor \( x^2 - 8x + 16 \)**:
   - This is a perfect square trinomial that factors into \( (x - 4)(x - 4) \), or \( (x - 4)^2 \).

### Step 2: Find the Least Common Denominator (LCD)
The denominators are \( (x + 7)(x - 4) \) and \( (x - 4)^2 \). The LCD is:

\[
(x + 7)(x - 4)^2.
\]

### Step 3: Rewrite Each Fraction with the LCD
1. For \( \frac{x + 2}{(x + 7)(x - 4)} \), multiply the numerator and denominator by \( x - 4 \):

\[
\frac{(x + 2)(x - 4)}{(x + 7)(x - 4)^2}.
\]

2. For \( \frac{x + 3}{(x - 4)^2} \), multiply the numerator and denominator by \( x + 7 \):

\[
\frac{(x + 3)(x + 7)}{(x + 7)(x - 4)^2}.
\]

### Step 4: Combine the Fractions
\[
\frac{(x + 2)(x - 4) + (x + 3)(x + 7)}{(x + 7)(x - 4)^2}.
\]

### Step 5: Expand and Simplify the Numerator
1. **Expand \( (x + 2)(x - 4) \)**:
\[
(x + 2)(x - 4) = x^2 - 4x + 2x - 8 = x^2 - 2x - 8.
\]

2. **Expand \( (x + 3)(x + 7) \)**:
\[
(x + 3)(x + 7) = x^2 + 7x + 3x + 21 = x^2 + 10x + 21.
\]

3. **Add the numerators**:
\[
x^2 - 2x - 8 + x^2 + 10x + 21 = 2x^2 + 8x + 13.
\]

### Step 6: Final Answer
The sum of the two fractions is:

\[
\frac{2x^2 + 8x + 13}{(x + 7)(x - 4)^2}.
\]

Would you like me to provide more details or clarify any step? 

Here are some related questions you might find helpful:

1. How do you factor quadratic expressions?
2. How do you determine the least common denominator for rational expressions?
3. What are the rules for adding and subtracting rational expressions?
4. How do you simplify complex fractions?
5. What techniques can be used to expand and simplify polynomial expressions?

**Tip:** Always factor expressions completely before finding a common denominator to make the process easier!

Perform the indicated operation: (x + 2) / (x^2 + 3x - 28) + (x + 3) / (x^2 - 8x + 16).

Learn how to add rational expressions by factoring quadratics and finding the least common denominator (LCD). This example demonstrates adding (x + 2)/(x^2 + 3x - 28) + (x + 3)/(x^2 - 8x + 16) with step-by-step solutions. Suitable for students in Grades 9-12.

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