1. **State the problem:** Solve the equation $$4^x - 3(2^x) = -2$$ for $x$. 2. **Rewrite the equation using properties of exponents:** Note that $$4^x = (2^2)^x = 2^{2x} = (2^x)^2$$. 3. **Substitute:** Let $$y = 2^x$$, so the equation becomes $$y^2 - 3y = -2$$. 4. **Rewrite the equation:** Move all terms to one side: $$y^2 - 3y + 2 = 0$$. 5. **Factor the quadratic:** $$y^2 - 3y + 2 = (y - 1)(y - 2) = 0$$. 6. **Solve for $y$:** $$y - 1 = 0 \Rightarrow y = 1$$ $$y - 2 = 0 \Rightarrow y = 2$$. 7. **Recall substitution:** Since $$y = 2^x$$, solve for $x$: - For $$2^x = 1$$, since $$2^0 = 1$$, $$x = 0$$. - For $$2^x = 2$$, since $$2^1 = 2$$, $$x = 1$$. 8. **Final answer:** $$x = 0$$ or $$x = 1$$.