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.

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.