The given function is:

\[
y = (x + 1)^2 e^x
\]

This is a combination of a polynomial \((x + 1)^2\) and the exponential function \(e^x\). Let's break it down and compute a few important properties of the function.

### 1. **First Derivative (dy/dx):**

To differentiate \(y\) with respect to \(x\), we'll use the product rule since it's a product of two functions: \(f(x) = (x + 1)^2\) and \(g(x) = e^x\). The product rule states:

\[
\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
\]

- \(f(x) = (x + 1)^2\)
- \(g(x) = e^x\)

Now, compute the derivatives of each:

- \(f'(x) = 2(x + 1)\)
- \(g'(x) = e^x\)

Now apply the product rule:

\[
\frac{dy}{dx} = 2(x + 1) e^x + (x + 1)^2 e^x
\]

Factor out \(e^x\):

\[
\frac{dy}{dx} = e^x \left[ 2(x + 1) + (x + 1)^2 \right]
\]

Simplify the expression inside the brackets:

\[
\frac{dy}{dx} = e^x \left[ 2(x + 1) + (x + 1)(x + 1) \right]
\]
\[
= e^x \left[ 2(x + 1) + (x^2 + 2x + 1) \right]
\]
\[
= e^x \left[ x^2 + 4x + 3 \right]
\]

Thus, the first derivative is:

\[
\frac{dy}{dx} = e^x (x^2 + 4x + 3)
\]

---

Would you like to explore the second derivative or other details about this function? 

Here are some additional related questions you may find useful:
1. How do we find the critical points of this function?
2. What is the second derivative \(\frac{d^2y}{dx^2}\)?
3. Can you graph this function?
4. How do you find the inflection points of this function?
5. What is the behavior of the function as \(x \to \infty\)?

**Tip:** When differentiating a product of functions, always use the product rule and simplify step by step to avoid mistakes.

Explore how to differentiate the function y = (x+1)^2 e^x using the product rule in calculus. This solution covers exponential functions, polynomials, and detailed step-by-step calculations to find the first derivative. Ideal for advanced high school or college-level calculus students.

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