To differentiate the expression \( A + (B_2x^2 + B_1x + B_0)e^{3x} \), we will use the sum rule, product rule, and the chain rule.

### Step-by-step differentiation:

1. \( A \) is a constant, so its derivative is zero:
   \[
   \frac{d}{dx}(A) = 0
   \]

2. For \( (B_2x^2 + B_1x + B_0)e^{3x} \), we will apply the product rule. The product rule states:
   \[
   \frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
   \]
   Here, \( f(x) = B_2x^2 + B_1x + B_0 \) and \( g(x) = e^{3x} \).

3. First, differentiate \( f(x) = B_2x^2 + B_1x + B_0 \):
   \[
   f'(x) = 2B_2x + B_1
   \]

4. Now, differentiate \( g(x) = e^{3x} \) using the chain rule. The chain rule tells us to multiply the derivative of the exponential by the derivative of the exponent:
   \[
   g'(x) = 3e^{3x}
   \]

5. Apply the product rule:
   \[
   \frac{d}{dx}[(B_2x^2 + B_1x + B_0)e^{3x}] = (2B_2x + B_1)e^{3x} + (B_2x^2 + B_1x + B_0) \cdot 3e^{3x}
   \]

6. Factor out \( e^{3x} \):
   \[
   \frac{d}{dx}[(B_2x^2 + B_1x + B_0)e^{3x}] = e^{3x}[(2B_2x + B_1) + 3(B_2x^2 + B_1x + B_0)]
   \]

7. Simplify the terms inside the brackets:
   \[
   = e^{3x}[(2B_2x + B_1) + 3B_2x^2 + 3B_1x + 3B_0]
   \]
   \[
   = e^{3x}(3B_2x^2 + (2B_2 + 3B_1)x + B_1 + 3B_0)
   \]

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

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

Here are 5 related questions:
1. How do you apply the product rule to more complex functions?
2. What is the significance of the chain rule in differentiating exponential functions?
3. How would you differentiate an expression involving multiple products and sums?
4. How can you differentiate a logarithmic function combined with a polynomial?
5. What happens when you differentiate \( e^{kx} \) for any constant \( k \)?

**Tip:** When differentiating products of functions, always carefully apply the product rule to avoid mistakes.

Learn how to differentiate the expression A + (B2x^2 + B1x + B0)e^3x step by step. This solution covers the product rule and chain rule, helping you master differentiation of polynomials and exponential functions. Ideal for students in Grades 11-12 or introductory calculus courses.

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