Let's simplify the given expression step by step.

We have:

\[
\frac{6x - 2}{2x + 14} \times \frac{x + 7}{12x - 4}
\]

### Step 1: Factor the numerators and denominators

1. **Factor the numerator \( 6x - 2 \):**

\[
6x - 2 = 2(3x - 1)
\]

2. **Factor the denominator \( 2x + 14 \):**

\[
2x + 14 = 2(x + 7)
\]

3. **Factor the numerator \( x + 7 \):**

\[
x + 7 \text{ is already factored.}
\]

4. **Factor the denominator \( 12x - 4 \):**

\[
12x - 4 = 4(3x - 1)
\]

### Step 2: Substitute the factored terms into the expression

\[
\frac{2(3x - 1)}{2(x + 7)} \times \frac{x + 7}{4(3x - 1)}
\]

### Step 3: Cancel common factors

1. Cancel the common factor of \( 2 \) in the first fraction.
2. Cancel the common factor of \( x + 7 \) in both fractions.
3. Cancel the common factor of \( 3x - 1 \) in both fractions.

After cancellation, we are left with:

\[
\frac{1}{4}
\]

Thus, the simplified form of the given expression is:

\[
\boxed{\frac{1}{4}}
\]

Would you like more details, or do you have any questions?

Here are some related questions:
1. How do you factor binomials with a common factor?
2. How do you identify common factors in rational expressions?
3. What happens when terms completely cancel out in a fraction?
4. How do you simplify expressions involving multiple fractions?
5. Can you explain how to simplify a more complex rational expression?

**Tip:** Always factor each term as much as possible when simplifying rational expressions to spot common factors easily.

Learn how to simplify the rational expression 6x-2/2x+14 * x+7/12x-4. This step-by-step solution involves factoring common terms and canceling out common factors. Ideal for students in Grades 8-10 looking to strengthen their algebra skills.

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