We are asked to find the value of:

$\frac{3x + 1}{x - 4} - \frac{x + 9}{x - 4}$

Since both terms have the same denominator $x - 4$, we can subtract the numerators directly while keeping the denominator the same.

Step 1: Subtract the numerators

$\frac{(3x + 1) - (x + 9)}{x - 4}$

Simplify the numerator: $(3x + 1) - (x + 9) = 3x + 1 - x - 9 = 2x - 8$

Step 2: Write the result

Now, the expression becomes: $\frac{2x - 8}{x - 4}$

Step 3: Factor the numerator (if possible)

$\frac{2(x - 4)}{x - 4}$

Since $x - 4$ appears in both the numerator and the denominator, they cancel out: $2$

Thus, the value of the expression is 2, which corresponds to option D.

Would you like further details or explanations? Here are some related questions to deepen your understanding:

1. How do you simplify fractions that have the same denominator?
2. Can you describe how factoring helps in simplifying algebraic expressions?
3. What happens when the numerator and denominator of a fraction share a common factor?
4. What should you watch out for when simplifying expressions like this?
5. How would the solution change if the denominators were different?

Tip: Always look for common factors in the numerator and denominator to simplify fractions quickly!