The correct form of the partial fraction decomposition of

$\frac{2x}{(x+3)(3x+1)}$

is determined based on the nature of the factors in the denominator. Since both $(x+3)$ and $(3x+1)$ are linear factors, the partial fraction decomposition takes the general form:

$\frac{2x}{(x+3)(3x+1)} = \frac{A}{x+3} + \frac{B}{3x+1},$

where $A$ and $B$ are constants to be determined.

From the choices given, Option B is correct, as it matches the standard form:

$\frac{A}{x+3} + \frac{B}{3x+1}.$

Let me know if you'd like further explanation on how to find $A$ and $B$ or any other related questions! Here are some follow-up questions for practice:

1. How do you determine the constants $A$ and $B$ in partial fraction decomposition?
2. Can you explain how the denominators impact the decomposition format?
3. How would the process change if one factor was quadratic instead of linear?
4. Can you generalize the decomposition for higher-degree polynomials?
5. How would you handle repeating factors in the denominator?

Tip: Always check the degrees of the numerator and denominator before starting partial fraction decomposition. If the numerator’s degree is greater than or equal to the denominator’s, perform polynomial division first!