1. **State the problem:** Solve for the variable $w$ in the equation $$4 \left(\frac{-6w - 9}{27} \right) - 2 = 2.$$\n\n2. **Rewrite the equation:** The equation is $$4 \times \frac{-6w - 9}{27} - 2 = 2.$$\n\n3. **Isolate the fraction term:** Add 2 to both sides to get rid of the $-2$ on the left:\n$$4 \times \frac{-6w - 9}{27} = 2 + 2 = 4.$$\n\n4. **Divide both sides by 4:**\n$$\frac{-6w - 9}{27} = \frac{4}{4} = 1.$$\n\n5. **Multiply both sides by 27 to clear the denominator:**\n$$-6w - 9 = 27 \times 1 = 27.$$\n\n6. **Add 9 to both sides:**\n$$-6w = 27 + 9 = 36.$$\n\n7. **Divide both sides by -6 to solve for $w$:**\n$$w = \frac{36}{-6} = -6.$$\n\n**Final answer:** $$w = -6.$$