Let’s break down and solve this problem step by step.

We are given the rational expression:

$\frac{\frac{2x^2 - 5x - 3}{3x^2 + 14x + 5}}{\frac{4x - 12}{x + 5}}$

When this is reciprocated (flipped), it becomes:

$\frac{2x^2 - 5x - 3}{3x^2 + 14x + 5} \cdot \frac{x + 5}{4x - 12}.$

Step 1: Factor all expressions

1. Numerator $2x^2 - 5x - 3$

Factor $2x^2 - 5x - 3$: $2x^2 - 5x - 3 = (2x + 1)(x - 3).$

2. Denominator $3x^2 + 14x + 5$

Factor $3x^2 + 14x + 5$: $3x^2 + 14x + 5 = (3x + 1)(x + 5).$

3. Denominator $4x - 12$

Factor $4x - 12$: $4x - 12 = 4(x - 3).$

Step 2: Write the reciprocated expression with factored forms

The expression now becomes: $\frac{(2x + 1)(x - 3)}{(3x + 1)(x + 5)} \cdot \frac{x + 5}{4(x - 3)}.$

Step 3: Simplify

1. Cancel $x + 5$ from numerator and denominator.
2. Cancel $x - 3$ from numerator and denominator.

The simplified expression is: $\frac{2x + 1}{4(3x + 1)}.$

Step 4: Match with the options

The factored form is: $\boxed{\text{a. } \frac{2x + 1}{4(3x - 1)}}.$

Do you have any questions or want a further explanation?

5 Related Questions:

1. How do you identify terms to factor in a quadratic expression?
2. What is the rule for reciprocating and simplifying rational expressions?
3. How do you handle complex fractions involving division and multiplication?
4. Why is it important to cancel common factors in rational expressions?
5. How do you verify the correctness of a simplified rational expression?

Tip:

Always double-check your factorizations, as small errors in factoring can lead to incorrect simplifications.