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The problem given is a partial differential equation (PDE) involving \( T(x,t) \), and we are tasked with finding \( V(x) \) such that the substitution \( T(x,t) = V(x) + \phi(x,t) \) turns the equation into a homogeneous PDE in \( \phi(x,t) \).

The given equation is:
\[
\frac{\partial}{\partial x} \left( \frac{1}{1 + 2x^2} \frac{\partial T}{\partial x} \right) = 8 + \frac{\partial T}{\partial t} + \frac{\partial^2 T}{\partial t^2}.
\]

To simplify this equation by substituting \( T(x,t) = V(x) + \phi(x,t) \), we first need to find \( V(x) \) such that the non-homogeneous terms vanish, i.e., the equation becomes homogeneous in terms of \( \phi(x,t) \).

### Substitution into the PDE

1. Substituting \( T(x,t) = V(x) + \phi(x,t) \), we get:
   \[
   \frac{\partial}{\partial x} \left( \frac{1}{1 + 2x^2} \frac{\partial}{\partial x} (V(x) + \phi(x,t)) \right) = 8 + \frac{\partial}{\partial t} (\phi(x,t)) + \frac{\partial^2}{\partial t^2} (\phi(x,t)).
   \]
2. This simplifies to:
   \[
   \frac{\partial}{\partial x} \left( \frac{1}{1 + 2x^2} \frac{\partial V(x)}{\partial x} \right) + \frac{\partial}{\partial x} \left( \frac{1}{1 + 2x^2} \frac{\partial \phi(x,t)}{\partial x} \right) = 8 + \frac{\partial \phi(x,t)}{\partial t} + \frac{\partial^2 \phi(x,t)}{\partial t^2}.
   \]
3. To achieve a homogeneous equation in \( \phi(x,t) \), the terms involving only \( V(x) \) must satisfy:
   \[
   \frac{\partial}{\partial x} \left( \frac{1}{1 + 2x^2} \frac{\partial V(x)}{\partial x} \right) = 8.
   \]

### Solving for \( V(x) \)

We now solve the equation:
\[
\frac{\partial}{\partial x} \left( \frac{1}{1 + 2x^2} \frac{\partial V(x)}{\partial x} \right) = 8.
\]

1. Integrate both sides with respect to \( x \):
   \[
   \frac{1}{1 + 2x^2} \frac{\partial V(x)}{\partial x} = 8x + C_1,
   \]
   where \( C_1 \) is an integration constant.

2. Multiply both sides by \( 1 + 2x^2 \):
   \[
   \frac{\partial V(x)}{\partial x} = (8x + C_1)(1 + 2x^2).
   \]

3. Integrate again with respect to \( x \):
   \[
   V(x) = \int (8x + C_1)(1 + 2x^2) dx.
   \]

   Expanding the integrand:
   \[
   V(x) = \int (8x + C_1)(1 + 2x^2) dx = \int (8x + 16x^3 + C_1 + 2C_1x^2) dx.
   \]

4. Perform the integration:
   \[
   V(x) = 4x^2 + 4x^4 + C_1x + C_2,
   \]
   where \( C_2 \) is another integration constant.

Comparing this with the answer choices, the correct form for \( V(x) \) is:
\[
V(x) = 4x^2 + 4x^4.
\]

### Conclusion:
The correct option is:
\[
V(x) = 4x^2 + 4x^4.
\]

Would you like more details or have any questions?

### Related Questions:
1. What are the conditions for a partial differential equation to be homogeneous?
2. How does the method of substitution work in simplifying differential equations?
3. Can we always find a function \( V(x) \) to make a PDE homogeneous?
4. How would the solution change if the right-hand side of the equation were different?
5. What other techniques exist for solving inhomogeneous PDEs?

**Tip:** When solving PDEs, always check the boundary or initial conditions to ensure that your solution satisfies the full problem setup.

Consider the inhomogeneous partial differential equation in T(x,t) given by: ∂/∂x((1/(1 + 2x^2)) ∂T/∂x) = 8 + ∂T/∂t + ∂^2T/∂t^2. Choose V(x) such that the substitution T(x,t) = V(x) + ϕ(x,t) gives rise to a homogeneous PDE in ϕ(x,t).

Learn how to solve an inhomogeneous partial differential equation by finding a function V(x) such that the equation becomes homogeneous in terms of ϕ(x,t). Detailed steps cover PDE manipulation, integration, and solution derivation for university-level students.

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