The expression shown in the image is:

$\frac{3}{x^2 + x - 2} - \frac{2}{x^2 + 3x + 2}$

To subtract these two rational expressions, we first need a common denominator.

Step 1: Factor the denominators

1. The first denominator $x^2 + x - 2$ can be factored as: $x^2 + x - 2 = (x - 1)(x + 2)$

2. The second denominator $x^2 + 3x + 2$ can be factored as: $x^2 + 3x + 2 = (x + 1)(x + 2)$

Step 2: Find the least common denominator (LCD)

The LCD is the product of all unique factors from both denominators, which is: $(x - 1)(x + 2)(x + 1)$

Step 3: Rewrite each fraction with the common denominator

1. Rewrite $\frac{3}{(x - 1)(x + 2)}$ with the LCD: $\frac{3}{(x - 1)(x + 2)} \cdot \frac{(x + 1)}{(x + 1)} = \frac{3(x + 1)}{(x - 1)(x + 1)(x + 2)}$

2. Rewrite $\frac{2}{(x + 1)(x + 2)}$ with the LCD: $\frac{2}{(x + 1)(x + 2)} \cdot \frac{(x - 1)}{(x - 1)} = \frac{2(x - 1)}{(x - 1)(x + 1)(x + 2)}$

Step 4: Subtract the numerators

Now, subtract the two fractions: $\frac{3(x + 1)}{(x - 1)(x + 1)(x + 2)} - \frac{2(x - 1)}{(x - 1)(x + 1)(x + 2)}$ Since they have the same denominator, subtract the numerators: $\frac{3(x + 1) - 2(x - 1)}{(x - 1)(x + 1)(x + 2)}$

Step 5: Simplify the numerator

Expand the terms in the numerator: $3(x + 1) = 3x + 3$ $2(x - 1) = 2x - 2$ Now subtract: $(3x + 3) - (2x - 2) = 3x + 3 - 2x + 2 = x + 5$

Final Answer

The simplified result is: $\frac{x + 5}{(x - 1)(x + 1)(x + 2)}$

Let me know if you'd like more details or have any questions!

Here are 5 related questions:

1. How do you factor quadratic expressions in general?
2. How do you find the least common denominator (LCD) for adding or subtracting rational expressions?
3. How would this process change if the expression was a sum instead of a difference?
4. What happens if one of the denominators cannot be factored?
5. How can you check the correctness of your solution for rational expressions?

Tip: Always factor denominators first when adding or subtracting rational expressions. It simplifies the process of finding a common denominator!