1. **State the problem:** Solve for $x$ in the equation $$-4x + 2(x + 4) = -7x - (x + 1)$$. 2. **Apply the distributive property:** Multiply through the parentheses: $$-4x + 2 \cdot x + 2 \cdot 4 = -7x - 1 \cdot x - 1 \cdot 1$$ which simplifies to $$-4x + 2x + 8 = -7x - x - 1$$. 3. **Combine like terms on each side:** Left side: $$-4x + 2x = -2x$$ Right side: $$-7x - x = -8x$$ So the equation becomes $$-2x + 8 = -8x - 1$$. 4. **Isolate variable terms on one side:** Add $8x$ to both sides: $$-2x + 8 + 8x = -8x - 1 + 8x$$ which is $$(-2x + \cancel{8x}) + 8 = \cancel{-8x} - 1 + \cancel{8x}$$ $$6x + 8 = -1$$. 5. **Isolate the constant term:** Subtract 8 from both sides: $$6x + 8 - 8 = -1 - 8$$ $$6x = -9$$. 6. **Solve for $x$ by dividing both sides by 6:** $$x = \frac{-9}{6}$$ Simplify the fraction by dividing numerator and denominator by 3: $$x = \frac{\cancel{-9}^{3}}{\cancel{6}^{2}} = -\frac{3}{2}$$. **Final answer:** $$x = -\frac{3}{2}$$.