To find \( x \in \mathbb{Z} \) such that \( P \in \mathbb{Z} \), you need the expression:

\[ P = \frac{2x^3 - x^2 + 3x + 4}{3x + 1} \]

to be an integer.

For \( P \) to be an integer, the numerator must be divisible by the denominator. Perform the polynomial long division to see when the remainder is zero.

1. **Divide:**
Divide the leading term of the numerator by the leading term of the denominator:
\(\frac{2x^3}{3x} = \frac{2}{3}x^2\).

2. **Multiply and Subtract:**
Multiply \(\frac{2}{3}x^2\) by \(3x + 1\) and subtract the result from the numerator.

3. **Continue:**
Continue dividing, multiplying, and subtracting until you get a remainder of zero or a division is not possible.

4. **Calculate Remainder:**
Check when the remainder equals zero. Solve the resulting equations or conditions to find integer values of \(x\).

Proceed with these steps using algebraic manipulation or check specific integer values of \(x\) to see where the expression results in an integer value of \(P\).

### Trial Method

Try specific integer values such as \(x = 0, 1, -1, 2, -2\) etc., to see when \( P \) is an integer by substituting straight into the original expression.

This approach often works well with smaller numbers and is reliant on identifying cancellation in the algebraic expression.

### Final Answer

By calculating or checking, you may find possible values of \(x\) that ensure \(P\) is an integer. Typically, finding one solution is reasonable due to the equation's cubic nature, and you can test around that point for more.

In practice, this method is facilitated greatly by the exact evaluation or specific manipulation techniques that reduce the remainder efficiently.