1. **State the problem:** We need to find the value of $k$ such that $x + 2$ is a factor of the polynomial $$x^3 - 2kx^2 + 6x - 4.$$\n\n2. **Recall the Factor Theorem:** If $x + 2$ is a factor, then substituting $x = -2$ into the polynomial should give zero.\n\n3. **Apply the Factor Theorem:** Substitute $x = -2$ into the polynomial:\n$$(-2)^3 - 2k(-2)^2 + 6(-2) - 4 = 0.$$\n\n4. **Simplify the expression:**\n$$-8 - 2k(4) - 12 - 4 = 0,$$\nwhich simplifies to\n$$-8 - 8k - 12 - 4 = 0.$$\n\n5. **Combine like terms:**\n$$-24 - 8k = 0.$$\n\n6. **Solve for $k$:**\n$$-8k = 24 \implies k = -3.$$\n\n**Final answer:** The value of $k$ is $-3$.