To solve the differential equation \( y' + 2y = x^2 + 2x \), we'll use the method of integrating factors.

### Step 1: Identify the integrating factor
The equation is in the standard linear form:
\[
y' + P(x)y = Q(x)
\]
where \( P(x) = 2 \) and \( Q(x) = x^2 + 2x \).

The integrating factor \( \mu(x) \) is given by:
\[
\mu(x) = e^{\int P(x) \, dx} = e^{\int 2 \, dx} = e^{2x}
\]

### Step 2: Multiply the equation by the integrating factor
Multiply the entire differential equation by \( e^{2x} \):
\[
e^{2x}y' + 2e^{2x}y = (x^2 + 2x)e^{2x}
\]

### Step 3: Recognize the left side as a derivative
The left side of the equation is now the derivative of \( y \) multiplied by the integrating factor:
\[
\frac{d}{dx}\left( e^{2x}y \right) = (x^2 + 2x)e^{2x}
\]

### Step 4: Integrate both sides
Integrate both sides with respect to \( x \):
\[
e^{2x}y = \int (x^2 + 2x)e^{2x} \, dx
\]

To solve the integral on the right side, we can use integration by parts. Let's solve it:

Let \( u = x^2 + 2x \) and \( dv = e^{2x} dx \). Then:
\[
du = (2x + 2)dx, \quad v = \frac{1}{2}e^{2x}
\]

So, using integration by parts \( \int u \, dv = uv - \int v \, du \):
\[
\int (x^2 + 2x)e^{2x} dx = \frac{1}{2}(x^2 + 2x)e^{2x} - \frac{1}{2} \int (2x + 2)e^{2x} dx
\]

For the remaining integral \( \int (2x + 2)e^{2x} dx \), we again apply integration by parts:
\[
\int (2x + 2)e^{2x} dx = \frac{1}{2}(2x + 2)e^{2x} - \int e^{2x} dx
\]
\[
= (x + 1)e^{2x} - \frac{1}{2}e^{2x}
\]

Substituting back:
\[
\int (x^2 + 2x)e^{2x} dx = \frac{1}{2}(x^2 + 2x)e^{2x} - \left[(x + 1)e^{2x} - \frac{1}{2}e^{2x}\right]
\]
\[
= \frac{1}{2}x^2e^{2x}
\]

### Step 5: Solve for \( y(x) \)
Finally, solve for \( y(x) \) by dividing both sides by \( e^{2x} \):
\[
y(x) = \frac{1}{2}x^2 + C e^{-2x}
\]
where \( C \) is the constant of integration.

So, the general solution is:
\[
\boxed{y(x) = \frac{1}{2}x^2 + Ce^{-2x}}
\]

Would you like more details or have any questions?

### Related Questions
1. How do you determine the integrating factor for a first-order linear differential equation?
2. What is the method of integrating factors used for in solving differential equations?
3. How do you perform integration by parts?
4. What are some common mistakes to avoid when solving linear differential equations?
5. How can you verify the solution to a differential equation?

### Tip:
When using the method of integrating factors, always check that the differential equation is in the correct linear form before proceeding.

Learn how to solve the differential equation y' + 2y = x^2 + 2x using the method of integrating factors and integration by parts. This problem involves advanced concepts in differential equations and is suitable for advanced high school students or undergraduates.

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