To derive the function \( A + (B_2x^2 + B_1x + B_0)e^{3x} \), we apply the sum rule and the product rule. Here's the detailed process:

### Given:
\[
f(x) = A + (B_2x^2 + B_1x + B_0)e^{3x}
\]

Where \( A \), \( B_2 \), \( B_1 \), and \( B_0 \) are constants.

### Step 1: Derivative of the first term \( A \)
Since \( A \) is a constant, its derivative is:
\[
\frac{d}{dx}(A) = 0
\]

### Step 2: Derivative of the second term \( (B_2x^2 + B_1x + B_0)e^{3x} \)

This is a product of two functions: 
- The first function is \( (B_2x^2 + B_1x + B_0) \)
- The second function is \( e^{3x} \)

We apply the **product rule**:
\[
\frac{d}{dx}(uv) = u'v + uv'
\]
Where:
- \( u = B_2x^2 + B_1x + B_0 \)
- \( v = e^{3x} \)

### Step 2a: Derivative of \( u \)
\[
u = B_2x^2 + B_1x + B_0
\]
\[
u' = \frac{d}{dx}(B_2x^2 + B_1x + B_0) = 2B_2x + B_1
\]

### Step 2b: Derivative of \( v \)
\[
v = e^{3x}
\]
\[
v' = \frac{d}{dx}(e^{3x}) = 3e^{3x}
\]

### Step 3: Apply the product rule
Using the product rule:
\[
\frac{d}{dx}\left((B_2x^2 + B_1x + B_0)e^{3x}\right) = \left(2B_2x + B_1\right)e^{3x} + (B_2x^2 + B_1x + B_0)\cdot 3e^{3x}
\]

### Step 4: Simplify
Factor out \( e^{3x} \):
\[
= \left[\left(2B_2x + B_1\right) + 3(B_2x^2 + B_1x + B_0)\right]e^{3x}
\]
Now expand the terms in the brackets:
\[
= \left[2B_2x + B_1 + 3B_2x^2 + 3B_1x + 3B_0\right]e^{3x}
\]
\[
= \left(3B_2x^2 + (2B_2 + 3B_1)x + (B_1 + 3B_0)\right)e^{3x}
\]

### Final derivative:
\[
\frac{d}{dx} \left(A + (B_2x^2 + B_1x + B_0)e^{3x}\right) = \left(3B_2x^2 + (2B_2 + 3B_1)x + (B_1 + 3B_0)\right)e^{3x}
\]

### Summary:
The derivative of the given expression is:
\[
\boxed{\left(3B_2x^2 + (2B_2 + 3B_1)x + (B_1 + 3B_0)\right)e^{3x}}
\]

Would you like further clarification or more details?

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Here are 5 related questions you might explore:
1. How do you apply the chain rule to a product involving exponential functions?
2. What is the significance of constants like \( B_2 \), \( B_1 \), and \( B_0 \) in derivative results?
3. How do higher-order derivatives affect expressions with polynomial and exponential combinations?
4. Can you derive higher powers of exponential functions like \( e^{nx} \)?
5. How do product rule and chain rule work together in complex functions?

**Tip:** When differentiating, always handle constants carefully, as their derivatives often simplify calculations!

Learn how to derive the function A + (B2x^2 + B1x + B0)e^3x using the product rule and sum rule. This detailed solution covers key differentiation concepts, including exponential functions and polynomials. Suitable for students in Grades 11-12.

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