The problem in the image asks us to calculate $f''(2)$ for the function:

$f(x) = x^3 + e^x(x^2 + x + 1)$

Steps to solve this:

1. Differentiate $f(x)$ to find $f'(x)$:

   • Apply the product rule to the second term $e^x(x^2 + x + 1)$, as it is a product of two functions.
   • The derivative of $x^3$ is straightforward.

2. Differentiate again to find $f''(x)$.

3. Substitute $x = 2$ into $f''(x)$ to calculate $f''(2)$.

Let's start with the differentiation. I'll compute this now.The second derivative $f''(x)$ is:

$f''(x) = 6x + 2(2x + 1)e^x + (x^2 + x + 1)e^x + 2e^x$

Now, substituting $x = 2$, we get:

$f''(2) = 12 + 19e^2$

This is the exact expression for $f''(2)$.

If you need a more detailed explanation or have any further questions, feel free to ask! Here are some related questions:

1. How does the product rule work in calculus?
2. Why is the second derivative important in analyzing functions?
3. What are some common applications of second derivatives in real-world problems?
4. How do exponential functions affect the behavior of derivatives?
5. Can we approximate $e^2$ to get a numerical value for $f''(2)$?

Tip: When working with exponential functions, it's often helpful to keep results in symbolic form to avoid approximation errors until the final step.