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The expression in the image involves adding two rational functions. The given expression is:

\[
\frac{x^2 - 3x - 18}{2x^2 + 11x + 15} + \frac{x^2 + 9x - 22}{2x^2 + x - 10}
\]

To solve this, we will:

1. Factor both denominators and numerators where possible.
2. Find the least common denominator (LCD) of the two fractions.
3. Combine the fractions by expressing them with the LCD.
4. Simplify the resulting expression.

Let’s begin with the factorization.

### Step 1: Factor the expressions
- \(x^2 - 3x - 18\): Factor this quadratic expression.
- \(2x^2 + 11x + 15\): Factor this quadratic expression.
- \(x^2 + 9x - 22\): Factor this quadratic expression.
- \(2x^2 + x - 10\): Factor this quadratic expression.

I'll compute these factorizations now.Here are the factorizations of the expressions:

- \( x^2 - 3x - 18 = (x - 6)(x + 3) \)
- \( 2x^2 + 11x + 15 = (x + 3)(2x + 5) \)
- \( x^2 + 9x - 22 = (x - 2)(x + 11) \)
- \( 2x^2 + x - 10 = (x - 2)(2x + 5) \)

### Step 2: Find the least common denominator (LCD)
The denominators are:
\[
(x + 3)(2x + 5) \quad \text{and} \quad (x - 2)(2x + 5)
\]
The LCD is the product of all unique factors:
\[
(x + 3)(x - 2)(2x + 5)
\]

### Step 3: Combine the fractions
Now, we'll rewrite both fractions with the LCD and then combine them.

\[
\frac{(x - 6)(x + 3)}{(x + 3)(2x + 5)} + \frac{(x - 2)(x + 11)}{(x - 2)(2x + 5)}
\]

I'll now simplify the sum.After combining the fractions, the resulting expression is:

\[
\frac{2x^3 + 7x^2 - 7x - 30}{(x - 2)(x + 3)(2x + 5)}
\]

This is the simplified form of the sum of the two rational functions.

Would you like a further breakdown of how the terms were combined, or do you have any specific questions?

Here are 5 related questions you can explore:
1. How do you find the least common denominator of rational expressions?
2. What are the steps to factor quadratic expressions?
3. How do you simplify the sum of two fractions with different denominators?
4. Can you explain the process of expanding polynomials?
5. How does factoring help in simplifying complex algebraic fractions?

**Tip:** Always factor the expressions in both the numerator and denominator before attempting to combine fractions—it simplifies the process!

Simplify the expression: (x^2 - 3x - 18) / (2x^2 + 11x + 15) + (x^2 + 9x - 22) / (2x^2 + x - 10).

Learn how to simplify the expression (x^2 - 3x - 18) / (2x^2 + 11x + 15) + (x^2 + 9x - 22) / (2x^2 + x - 10) by factoring and finding the least common denominator. This problem covers algebraic concepts suitable for students in Grades 9-11.

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