We are asked to solve the integral:

\[
\int \frac{x^2 + 2x}{(x+1)^2} \, dx
\]

### Step 1: Simplifying the integrand
First, simplify the expression by dividing the numerator term by term:

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

Now, rewrite the expression in a form that is easier to integrate. Perform long division on the polynomial, or use partial fraction decomposition. Begin by performing polynomial long division.

### Step 2: Polynomial long division
Divide \(x^2 + 2x\) by \((x+1)^2\):

1. Divide \(x^2\) by \(x^2\) (the highest degree term):
   \[
   \frac{x^2}{(x+1)^2} = 1
   \]
2. Subtract \((x^2 + 2x)\) from \((x+1)^2 \cdot 1\), leaving a remainder of \(0\).

The quotient is:

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

### Step 3: Solve the integral
Now, the integral becomes simpler:

\[
\int 1 \, dx = x + C
\]

### Final Answer:
\[
\int \frac{x^2 + 2x}{(x+1)^2} \, dx = x + C
\]

Would you like further clarification, or are there any details you'd like to dive deeper into?

Here are five related questions you might find helpful:
1. How does polynomial long division work in integrals?
2. How can partial fraction decomposition simplify integrals?
3. What are the applications of integrals in real-world problems?
4. How can you check the result of an indefinite integral?
5. What is the significance of the constant of integration?

**Tip:** Always simplify the integrand as much as possible before attempting to integrate—it often leads to easier solutions.

Discover how to solve the integral ∫(x²+2x)/(x+1)² dx using polynomial long division. This detailed guide simplifies the integrand and provides an easy-to-follow solution, perfect for calculus students in advanced high school or undergraduate courses.

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