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The question is asking which of the given differential equations is of the variable separable type. In variable separable equations, the differential equation can be rewritten such that all terms involving \( x \) are on one side and all terms involving \( y \) are on the other side. Let's examine each equation to check for separability.

### Option A:
\[
4(x^2 + 1) dx + 2x \tan y \, dx = (x^2 + 1) \sec^2 y \, dy
\]
This equation has mixed terms involving both \( x \) and \( y \) on both sides of the equation, and it cannot be easily separated into functions of \( x \) and \( y \) alone. **Not separable**.

### Option B:
\[
dy - y^3 e^{2x} \sin x \, dx - y \, dx = 0
\]
This equation can be rewritten as:
\[
dy = y^3 e^{2x} \sin x \, dx + y \, dx
\]
This is not separable because \( y \) and \( x \) are multiplied together, and it cannot be easily rearranged into a separable form. **Not separable**.

### Option C:
\[
\left[ (y - 2y^3) e^{xy} dx + x e^{xy} dy \right] + (6y^2 - 2) y \, dy = 0
\]
This equation has complex terms involving both \( x \) and \( y \) inside exponentials and products, making it hard to separate into terms solely involving \( x \) and \( y \). **Not separable**.

### Option D:
\[
(xy + x) dx = (x^2 y^2 + x^2 + y^2 + 1) dy
\]
This equation can be separated because the terms involving \( x \) and \( y \) can be divided and moved to opposite sides:
\[
\frac{(xy + x)}{(x^2 y^2 + x^2 + y^2 + 1)} = \frac{dy}{dx}
\]
While it appears complicated, it is possible to manipulate this equation into a separable form. **Separable**.

### Answer:
**Option D** is the variable separable differential equation.

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Would you like further details on how to separate the variables for Option D or explanations of any specific steps? Here are some related questions you might find helpful:

1. What are the steps to solve a separable differential equation?
2. How can nonlinear terms affect the separability of a differential equation?
3. Can all first-order differential equations be made separable?
4. What are common strategies to simplify complex separable equations?
5. How can exponential and trigonometric functions impact the separation process?

**Tip**: When trying to identify separable equations, always look for a way to rewrite the equation as a product of functions of \(x\) and \(y\) only, each on opposite sides of the equation.

Which of the following differential equations is a variable separable type of differential equation? The options are: A. 4(x^2 + 1) dx + 2x tan y dx = (x^2 + 1) sec^2 y dy, B. dy - y^3 e^(2x) sin x dx - y dx = 0, C. [(y - 2y^3) e^(xy) dx + x e^(xy) dy] + (6y^2 - 2) y dy = 0, D. (xy + x) dx = (x^2 y^2 + x^2 + y^2 + 1) dy.

Identify the variable separable differential equation among given options. Explore the steps to recognize and solve such equations with algebraic techniques. Suitable for advanced high school and college students.

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