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To solve the problem, we need to find the derivative of the given function \( y = (2x^2 + 6x)(3x - 4) \) using the Product Rule. The Product Rule states that if \( y = uv \), where \( u \) and \( v \) are functions of \( x \), then the derivative \( y' \) is given by:

\[
y' = u'v + uv'
\]

### Step 1: Identify \( u \) and \( v \)
From the problem:
- \( u = 2x^2 + 6x \)
- \( v = 3x - 4 \)

### Step 2: Find the derivatives \( u' \) and \( v' \)
- \( u' = \frac{d}{dx}(2x^2 + 6x) \)
\[
u' = 4x + 6
\]
- \( v' = \frac{d}{dx}(3x - 4) \)
\[
v' = 3
\]

### Step 3: Apply the Product Rule
\[
y' = u'v + uv'
\]
Substitute \( u \), \( u' \), \( v \), and \( v' \) into the equation:
\[
y' = (4x + 6)(3x - 4) + (2x^2 + 6x)(3)
\]

### Step 4: Simplify each term
- First term: \( (4x + 6)(3x - 4) \)
\[
(4x + 6)(3x - 4) = 12x^2 - 16x + 18x - 24 = 12x^2 + 2x - 24
\]

- Second term: \( (2x^2 + 6x)(3) \)
\[
(2x^2 + 6x)(3) = 6x^2 + 18x
\]

### Step 5: Combine like terms
\[
y' = (12x^2 + 2x - 24) + (6x^2 + 18x) = 18x^2 + 20x - 24
\]

### Final Answer
The derivative \( y' \) is \( 18x^2 + 20x - 24 \).

Looking at the options provided, this corresponds to the following choice:

- \( \boxed{\text{Option C}} \)

Would you like further details on this solution or have any questions?

Here are some related questions you might find helpful:

1. What is the general form of the Product Rule, and when is it applicable?
2. How do you find the derivative of a polynomial function?
3. What are some common mistakes to avoid when using the Product Rule?
4. How does the Chain Rule differ from the Product Rule in calculus?
5. Can the Product Rule be extended to three or more functions?
6. How do you interpret the result of a derivative in a real-world context?
7. What is the significance of combining like terms after finding derivatives?
8. How does the choice of \( u \) and \( v \) affect the calculation in the Product Rule?

**Tip:** Always double-check your algebraic simplifications when applying the Product Rule to avoid errors in the final derivative.

Learn how to solve a derivative problem using the Product Rule in calculus. Step-by-step solution provided for finding the derivative of a complex function involving polynomial terms. Suitable for advanced high school and college students.

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