To find the simplified form of the given function:

$f(x) = \left(e^{x^2 + 2x} \cdot e^{-x - x^2}\right)$

we can use the property of exponents $e^a \cdot e^b = e^{a + b}$. Let's combine the exponents:

$f(x) = e^{(x^2 + 2x) + (-x - x^2)}$

Simplify the exponent:

$(x^2 + 2x) + (-x - x^2) = x^2 + 2x - x - x^2 = x$

So the function simplifies to:

$f(x) = e^x$

Therefore, the answer is:

$f(x) = e^x$

Would you like any further details or have any questions? Here are some related questions:

1. How do you combine exponents when multiplying exponential expressions?
2. What are the basic properties of exponents that are useful in simplification?
3. How can you simplify more complex exponentials involving multiple variables?
4. What is the derivative of $f(x) = e^x$?
5. How do exponential functions behave as $x$ approaches positive and negative infinity?
6. What are some common applications of exponential functions in real-world scenarios?
7. How do you integrate functions involving exponentials?
8. What is the inverse function of $f(x) = e^x$?

Tip: When working with exponential functions, always look for opportunities to simplify using properties of exponents, such as combining terms with the same base.