To determine the difference between the given rational expressions:

$\frac{9x}{x^2 - 9} - \frac{2x}{x^2 + 7x + 12},$

let's solve this step-by-step.

Step 1: Factorize the denominators

1. $x^2 - 9$ is a difference of squares: $x^2 - 9 = (x - 3)(x + 3).$
2. $x^2 + 7x + 12$ factors as: $x^2 + 7x + 12 = (x + 3)(x + 4).$

So, the problem becomes: $\frac{9x}{(x - 3)(x + 3)} - \frac{2x}{(x + 3)(x + 4)}.$

Step 2: Find the least common denominator (LCD)

The common denominator between the two fractions is: $\text{LCD} = (x - 3)(x + 3)(x + 4).$

Step 3: Rewrite each fraction with the LCD

1. For the first fraction: $\frac{9x}{(x - 3)(x + 3)} = \frac{9x(x + 4)}{(x - 3)(x + 3)(x + 4)}.$
2. For the second fraction: $\frac{2x}{(x + 3)(x + 4)} = \frac{2x(x - 3)}{(x - 3)(x + 3)(x + 4)}.$

Step 4: Combine the fractions

Combine the numerators over the common denominator: $\frac{9x(x + 4)}{(x - 3)(x + 3)(x + 4)} - \frac{2x(x - 3)}{(x - 3)(x + 3)(x + 4)}.$

The numerator becomes: $9x(x + 4) - 2x(x - 3).$

Simplify each term:

• $9x(x + 4) = 9x^2 + 36x$,
• $2x(x - 3) = 2x^2 - 6x$.

Now subtract: $(9x^2 + 36x) - (2x^2 - 6x).$

Simplify: $9x^2 + 36x - 2x^2 + 6x = 7x^2 + 42x.$

Step 5: Final expression

The result is: $\frac{7x^2 + 42x}{(x - 3)(x + 3)(x + 4)}.$

Step 6: Verify answer

Comparing with the options, the correct answer is:

$\boxed{\frac{7x^2 + 42x}{(x + 3)(x - 3)(x + 4)}, \, x \neq -3, \, x \neq 3, \, x \neq -4.}$

Do you want me to further explain any steps?

Here are 5 related questions for you to explore:

1. How do you find the least common denominator for rational expressions?
2. What is the general process for subtracting fractions with different denominators?
3. How do you factorize a quadratic trinomial like $x^2 + 7x + 12$?
4. How do you simplify a difference involving rational functions?
5. What is the domain of a rational function?

Tip: Always factorize the denominators completely before combining rational expressions.