Let's analyze and solve this problem.

Problem Breakdown:

We are given a rational expression:
$\frac{x^2 - 5x - 3}{3x^2 + 14x + 5} \div \frac{4x - 12}{x + 5}$ This is then reciprocated and simplified. The task is to find the factored form.

Step 1: Rewrite the division as multiplication

Division of fractions means multiplying by the reciprocal of the second fraction: $\frac{x^2 - 5x - 3}{3x^2 + 14x + 5} \div \frac{4x - 12}{x + 5} = \frac{x^2 - 5x - 3}{3x^2 + 14x + 5} \times \frac{x + 5}{4x - 12}$

Step 2: Factorize all components

1. Numerator of the first fraction: $x^2 - 5x - 3$
   This cannot be factored further (no simple roots). It remains as it is.

2. Denominator of the first fraction: $3x^2 + 14x + 5$
   Factorizing this quadratic:

   • Multiply $3 \times 5 = 15$
   • Find two numbers that multiply to 15 and add to 14: $15$ and $1$
   • Split the middle term:
     $3x^2 + 15x + x + 5 = (3x^2 + 15x) + (x + 5) = 3x(x + 5) + 1(x + 5) = (3x + 1)(x + 5)$ So, $3x^2 + 14x + 5 = (3x + 1)(x + 5)$.

3. Numerator of the second fraction: $x + 5$
   This is already factored.

4. Denominator of the second fraction: $4x - 12$
   Factor out 4:
   $4x - 12 = 4(x - 3)$

Step 3: Rewrite the expression

Substituting the factored components: $\frac{x^2 - 5x - 3}{(3x + 1)(x + 5)} \times \frac{x + 5}{4(x - 3)}$

Step 4: Cancel common terms

The $x + 5$ terms in the numerator and denominator cancel out. The simplified expression becomes: $\frac{x^2 - 5x - 3}{(3x + 1)} \times \frac{1}{4(x - 3)} = \frac{x^2 - 5x - 3}{4(3x + 1)(x - 3)}$

Step 5: Match the options

Now, let's rewrite this to match the given answer choices. The numerator $x^2 - 5x - 3$ remains unfactored. The denominator is $4(3x + 1)(x - 3)$.

The correct choice is: [ \boxed{d. \frac{2x + 1}{3x - 1}} \Let me know if you want details about additional factoring concepts or step-by-step verification?