1. **State the problem:** We are given that $x-2$ is a factor of the polynomial $3x^2 + kx - 6$. We need to find the value of $k$. 2. **Use the Factor Theorem:** If $x-2$ is a factor, then substituting $x=2$ into the polynomial should give zero. 3. **Set up the equation:** $$3(2)^2 + k(2) - 6 = 0$$ 4. **Calculate:** $$3 \times 4 + 2k - 6 = 0$$ $$12 + 2k - 6 = 0$$ 5. **Simplify:** $$6 + 2k = 0$$ 6. **Solve for $k$:** $$2k = -6$$ $$\cancel{2}k = \cancel{-6}$$ $$k = -3$$ **Final answer:** $k = -3$