The given differential equation is:

\[
\frac{dP}{dt} + 2tP = P - 2 + 4t
\]

### Step 1: Rewrite the equation
First, let's move all terms involving \(P\) to the left side and constant or independent terms to the right side:

\[
\frac{dP}{dt} + 2tP - P = 4t - 2
\]

Simplify the terms involving \(P\):

\[
\frac{dP}{dt} + (2t - 1)P = 4t - 2
\]

This is a first-order linear differential equation of the form \(\frac{dP}{dt} + a(t)P = b(t)\), where \(a(t) = 2t - 1\) and \(b(t) = 4t - 2\).

### Step 2: Find the integrating factor
The integrating factor \(\mu(t)\) is given by:

\[
\mu(t) = e^{\int a(t) \, dt} = e^{\int (2t - 1) \, dt}
\]

Let's calculate this integral:

\[
\mu(t) = e^{t^2 - t}
\]

### Step 3: Multiply the differential equation by the integrating factor
We multiply both sides of the differential equation by \(e^{t^2 - t}\):

\[
e^{t^2 - t} \frac{dP}{dt} + (2t - 1)e^{t^2 - t}P = (4t - 2)e^{t^2 - t}
\]

The left-hand side is the derivative of \(\mu(t)P\):

\[
\frac{d}{dt}\left(e^{t^2 - t}P\right) = (4t - 2)e^{t^2 - t}
\]

### Step 4: Integrate both sides with respect to \(t\)
Integrate both sides:

\[
e^{t^2 - t}P = \int (4t - 2)e^{t^2 - t} \, dt
\]

### Step 5: Solve the integral
Let's solve the integral on the right-hand side using substitution. Set \(u = t^2 - t\), then \(du = (2t - 1) dt\).

This means we can rewrite the integral as:

\[
\int (4t - 2)e^{t^2 - t} \, dt = 2 \int e^u \, du = 2e^u + C = 2e^{t^2 - t} + C
\]

### Step 6: Solve for \(P(t)\)
Finally, solve for \(P(t)\):

\[
e^{t^2 - t}P = 2e^{t^2 - t} + C
\]

Dividing by \(e^{t^2 - t}\) on both sides:

\[
P(t) = 2 + Ce^{-(t^2 - t)}
\]

### Final Solution:
The general solution to the differential equation is:

\[
P(t) = 2 + Ce^{-(t^2 - t)}
\]

where \(C\) is the constant of integration.

Would you like more details or have any other questions? Here are some related questions to consider:

1. What is the significance of the constant \(C\) in the solution?
2. How does the solution behave as \(t\) becomes very large or very small?
3. Can you solve for \(C\) if an initial condition \(P(t_0) = P_0\) is given?
4. How does the form of \(a(t)\) in the integrating factor affect the solution?
5. How would the solution change if the non-homogeneous term were different?

**Tip:** After finding the general solution to a differential equation, always check if it satisfies the original equation by substituting it back in.

Learn how to solve the first-order linear differential equation dP/dt + 2tP = P - 2 + 4t step-by-step. This problem involves using integrating factors and solving linear differential equations. Suitable for advanced high school students or anyone studying differential equations.

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