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The given differential equation is:

\[
\frac{dy}{dx} = xy + 4x + 2y + 8
\]

This is a first-order non-homogeneous linear differential equation. We can solve it by using an integrating factor.

### Step 1: Rearrange the equation

First, we rewrite the equation in the standard form:

\[
\frac{dy}{dx} - (x+2)y = 4x + 8
\]

### Step 2: Find the integrating factor

The integrating factor (IF) is given by:

\[
IF = e^{\int -(x+2)dx} = e^{-x^2/2 - 2x}
\]

### Step 3: Multiply the whole equation by the integrating factor

Multiplying both sides of the differential equation by the integrating factor:

\[
e^{-x^2/2 - 2x} \frac{dy}{dx} - (x+2) e^{-x^2/2 - 2x} y = (4x + 8) e^{-x^2/2 - 2x}
\]

The left-hand side becomes:

\[
\frac{d}{dx}\left( e^{-x^2/2 - 2x} y \right)
\]

### Step 4: Integrate both sides

Now, integrate both sides with respect to \(x\):

\[
\int \frac{d}{dx}\left( e^{-x^2/2 - 2x} y \right) dx = \int (4x + 8) e^{-x^2/2 - 2x} dx
\]

### Step 5: Solve for \(y\)

After integration, solve for \(y\) and include the constant of integration \(k\).

---

Would you like the detailed steps for integration, or have any specific questions? 

Here are some related questions you might explore:

1. What are the different methods for solving non-homogeneous differential equations?
2. How does an integrating factor work in solving linear differential equations?
3. What is the role of arbitrary constants in the solution of differential equations?
4. How do you handle non-homogeneous terms in differential equations?
5. Can this differential equation be solved using a series solution?

**Tip:** In solving linear differential equations, always ensure the equation is in the correct form before applying the integrating factor method.

Find a solution to dy/dx = xy + 4x + 2y + 8. If necessary, use k to denote an arbitrary constant.

Find the solution to the differential equation dy/dx = xy + 4x + 2y + 8 using the integrating factor method. This step-by-step solution covers essential concepts in differential equations, including integrating factors and handling non-homogeneous terms.

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