We are asked to add the rational expressions:

$\frac{3}{x^2 + 6x + 9} + \frac{2}{x^2 - 9}$

Step 1: Factor the denominators

First, factor the denominators, if possible:

1. $x^2 + 6x + 9$ is a perfect square trinomial: $x^2 + 6x + 9 = (x + 3)^2$

2. $x^2 - 9$ is a difference of squares: $x^2 - 9 = (x - 3)(x + 3)$

So, the original expression becomes:

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

Step 2: Find the least common denominator (LCD)

The LCD must include both $(x + 3)^2$ and $(x - 3)(x + 3)$. The least common denominator is:

$\text{LCD} = (x + 3)^2(x - 3)$

Step 3: Rewrite each fraction with the LCD

We need to adjust both fractions to have the same denominator:

• The first fraction: $\frac{3}{(x + 3)^2} = \frac{3(x - 3)}{(x + 3)^2(x - 3)}$

• The second fraction: $\frac{2}{(x - 3)(x + 3)} = \frac{2(x + 3)}{(x - 3)(x + 3)^2}$

Step 4: Add the numerators

Now, we can add the fractions, as they have the same denominator:

$\frac{3(x - 3)}{(x + 3)^2(x - 3)} + \frac{2(x + 3)}{(x + 3)^2(x - 3)} = \frac{3(x - 3) + 2(x + 3)}{(x + 3)^2(x - 3)}$

Step 5: Simplify the numerator

Distribute the terms in the numerator:

$3(x - 3) = 3x - 9$ $2(x + 3) = 2x + 6$

Now add them together:

$(3x - 9) + (2x + 6) = 5x - 3$

Step 6: Final expression

The final result is:

$\frac{5x - 3}{(x + 3)^2(x - 3)}$

This is the simplified sum of the two rational expressions.

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Would you like more details on any step, or have any questions?

Related Questions:

1. How do you subtract rational expressions?
2. Can you explain the factoring of perfect square trinomials in more detail?
3. What happens if one of the denominators is already factored?
4. How can you simplify complex rational expressions?
5. What if the numerators were more complicated—how would the process change?

Tip:

Always factor the denominators first when adding or subtracting rational expressions to make finding the LCD easier.