Question: Solve the following equation for x. Express your answer in the simplest form.
2(4x + 3) + 4 = -2(-4x - 6) - 2
1. **State the problem:** Solve the equation $$2(4x + 3) + 4 = -2(-4x - 6) - 2$$ for $$x$$.
2. **Apply the distributive property:**
$$2 \times 4x + 2 \times 3 + 4 = -2 \times (-4x) - 2 \times (-6) - 2$$
which simplifies to
$$8x + 6 + 4 = 8x + 12 - 2$$
3. **Combine like terms on each side:**
Left side: $$6 + 4 = 10$$
Right side: $$12 - 2 = 10$$
So the equation becomes
$$8x + 10 = 8x + 10$$
4. **Subtract $$8x$$ from both sides:**
$$8x + 10 - \cancel{8x} = 8x + 10 - \cancel{8x}$$
which simplifies to
$$10 = 10$$
5. **Interpret the result:**
Since $$10 = 10$$ is always true, the equation holds for all values of $$x$$.
**Final answer:** The solution is all real numbers, or $$x \in \mathbb{R}$$.