To divide $4x^3 - 3x^2 + x - 2$ by $x + 2$ using long division, we proceed as follows:

1. Set up the division: We are dividing $4x^3 - 3x^2 + x - 2$ by $x + 2$.

2. Divide the first term:

   • Divide the leading term of the dividend, $4x^3$, by the leading term of the divisor, $x$, which gives $4x^2$.
   • Write $4x^2$ above the division line.

3. Multiply and Subtract:

   • Multiply $4x^2$ by $x + 2$: $4x^2 \cdot x = 4x^3$ and $4x^2 \cdot 2 = 8x^2$, giving $4x^3 + 8x^2$.
   • Subtract this result from the original polynomial: $(4x^3 - 3x^2) - (4x^3 + 8x^2) = -11x^2$

4. Repeat the process:

   • Divide $-11x^2$ by $x$, which gives $-11x$.
   • Multiply $-11x$ by $x + 2$: $-11x \cdot x = -11x^2$ and $-11x \cdot 2 = -22x$, giving $-11x^2 - 22x$.
   • Subtract this result: $(-11x^2 + x) - (-11x^2 - 22x) = 23x$

5. Continue dividing:

   • Divide $23x$