$$ \frac{\left(\frac{14a}{a+2}\right)}{\left(\frac{7}{6a+12}\right)} $$

We are asked to simplify the expression:

\[
\frac{\left(\frac{14a}{a+2}\right)}{\left(\frac{7}{6a+12}\right)}
\]

### Step 1: Simplify the denominator
First, notice that the denominator of the second fraction can be factored:
\[
6a + 12 = 6(a + 2)
\]
Thus, the second fraction becomes:
\[
\frac{7}{6a + 12} = \frac{7}{6(a + 2)}
\]

### Step 2: Rewrite the entire expression
Now, the expression becomes:
\[
\frac{\left(\frac{14a}{a+2}\right)}{\left(\frac{7}{6(a+2)}\right)}
\]

### Step 3: Apply division of fractions
Dividing by a fraction is the same as multiplying by its reciprocal. So, we multiply by the reciprocal of the second fraction:
\[
\frac{14a}{a+2} \times \frac{6(a+2)}{7}
\]

### Step 4: Cancel common factors
Notice that $ a + 2 $ appears in both the numerator and denominator, so we can cancel those terms:
\[
\frac{14a}{\cancel{a+2}} \times \frac{6\cancel{(a+2)}}{7}
= \frac{14a \times 6}{7}
\]

### Step 5: Simplify
Now, simplify the expression:
\[
\frac{14a \times 6}{7} = \frac{84a}{7} = 12a
\]

Thus, the simplified expression is:

\[
\boxed{12a}
\]

Would you like further explanation, or do you have any questions?

### Related Questions:
1. How can you simplify rational expressions involving polynomials?
2. What are the steps to factor common terms in algebraic fractions?
3. How do you divide and multiply complex fractions?
4. What are the general rules for canceling terms in fractions?
5. How do you deal with expressions involving variables in both the numerator and denominator?

### Tip:
Always look for common factors between the numerator and denominator before multiplying or dividing fractions—it simplifies the process and reduces mistakes.

Learn how to simplify a complex rational expression involving fractions, using factoring and canceling common terms. This problem demonstrates division of fractions and how to simplify algebraic expressions step by step. Suitable for students in Grades 9-11.

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