Let's break down the problem and work through it step by step.

The problem is:

$\left( \frac{2}{3x - 2} - \frac{3}{x + 4} \right) \div \frac{7x - 14}{12x^2 + 7x - 10}$

We are asked to simplify this into a single fraction. Let's start with the following steps:

Step 1: Simplify the subtraction inside the parentheses

We need to subtract the two fractions:

$\frac{2}{3x - 2} - \frac{3}{x + 4}$

To subtract them, we need a common denominator. The common denominator of $(3x - 2)$ and $(x + 4)$ is their product:

$(3x - 2)(x + 4)$

Now rewrite each fraction with this denominator:

$\frac{2}{3x - 2} = \frac{2(x + 4)}{(3x - 2)(x + 4)}$ $\frac{3}{x + 4} = \frac{3(3x - 2)}{(3x - 2)(x + 4)}$

Now subtract the numerators:

$\frac{2(x + 4) - 3(3x - 2)}{(3x - 2)(x + 4)}$

Simplify the numerator:

$2(x + 4) = 2x + 8$ $3(3x - 2) = 9x - 6$

Now subtract them:

$(2x + 8) - (9x - 6) = 2x + 8 - 9x + 6 = -7x + 14$

So, we now have:

$\frac{-7x + 14}{(3x - 2)(x + 4)}$

Step 2: Simplify the division

Now, divide this result by the second fraction:

$\frac{-7x + 14}{(3x - 2)(x + 4)} \div \frac{7x - 14}{12x^2 + 7x - 10}$

Dividing by a fraction is the same as multiplying by its reciprocal:

$\frac{-7x + 14}{(3x - 2)(x + 4)} \times \frac{12x^2 + 7x - 10}{7x - 14}$

Factor where possible:

• $-7x + 14 = -7(x - 2)$
• $7x - 14 = 7(x - 2)$

Now the expression becomes:

$\frac{-7(x - 2)}{(3x - 2)(x + 4)} \times \frac{12x^2 + 7x - 10}{7(x - 2)}$

Notice that $(x - 2)$ cancels out from both the numerator and denominator:

$\frac{-7}{(3x - 2)(x + 4)} \times \frac{12x^2 + 7x - 10}{7}$

Cancel out the 7:

$\frac{-1}{(3x - 2)(x + 4)} \times (12x^2 + 7x - 10)$

Now multiply the numerators:

$\frac{-(12x^2 + 7x - 10)}{(3x - 2)(x + 4)}$

Thus, the simplified expression is:

$\boxed{\frac{-(12x^2 + 7x - 10)}{(3x - 2)(x + 4)}}$

Let me know if you would like further details or explanations.

Here are 5 questions you might explore further:

1. How can you find the least common denominator for more complex fractions?
2. What happens if there are higher-degree polynomials in the numerator?
3. Can this problem be solved using an alternative method, like partial fractions?
4. How do you approach factoring when polynomials are more difficult?
5. What is the significance of cancelling terms in simplifying rational expressions?

Tip: Always double-check for factoring opportunities in both the numerator and denominator before canceling terms in rational expressions.