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The problem asks for the degree 2 Taylor polynomial for the function \( e^{5x + 5} \) centered at \( x = 0 \).

### Step 1: The general formula for a Taylor polynomial
For a function \( f(x) \), the Taylor polynomial of degree 2 centered at \( x = 0 \) is given by:

\[
T_2(x) = f(0) + f'(0)x + \frac{f''(0)}{2}x^2
\]

### Step 2: Calculate \( f(x) = e^{5x + 5} \), its derivatives, and values at \( x = 0 \)
- \( f(x) = e^{5x + 5} = e^5 \cdot e^{5x} \)
- First derivative: \( f'(x) = 5e^{5x + 5} = 5e^5 \cdot e^{5x} \)
- Second derivative: \( f''(x) = 25e^{5x + 5} = 25e^5 \cdot e^{5x} \)

Now, evaluate at \( x = 0 \):
- \( f(0) = e^{5(0) + 5} = e^5 \)
- \( f'(0) = 5e^5 \)
- \( f''(0) = 25e^5 \)

### Step 3: Construct the degree 2 Taylor polynomial
Using the values from the derivatives:
\[
T_2(x) = e^5 + 5e^5x + \frac{25e^5}{2}x^2
\]

### Step 4: Compare with the options
The polynomial matches the second option:
\[
\frac{25}{2}e^5x^2 + 5e^5x + e^5
\]

This is the correct answer.

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Do you need further clarification or additional details? 

Here are five related questions you might find useful:
1. How do we derive the general formula for a Taylor polynomial?
2. What is the significance of the degree of a Taylor polynomial?
3. How do higher-order Taylor polynomials approximate functions more accurately?
4. Can the Taylor series be used to approximate functions around points other than \( x = 0 \)?
5. What happens to the remainder term in the Taylor expansion?

**Tip:** Always calculate the necessary derivatives carefully when constructing a Taylor series to avoid small errors.

Find the degree 2 Taylor polynomial for the function e^{5x+5} centered at x = 0.

Solve a degree 2 Taylor polynomial for the function e^{5x+5} centered at x = 0. This problem involves the calculation of derivatives and constructing a polynomial using Taylor series expansion. Suitable for college-level calculus students.

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