Let's solve the equation:

Given: $4((2x + 8) = 2x + 4)$

To proceed, it seems like there was a misinterpretation in the original equation due to the way it was written. I assume you intended to write:

$4 \cdot (2x + 8) = 2x + 4$

Let's solve it step-by-step.

1. Distribute the $4$ on the left side: $4 \cdot (2x + 8) = 2x + 4$ becomes: $8x + 32 = 2x + 4$

2. Move all terms involving $x$ to one side and constants to the other: $8x - 2x = 4 - 32$ Simplifying, we get: $6x = -28$

3. Divide both sides by $6$ to solve for $x$: $x = -\frac{28}{6}$ Simplifying further: $x = -\frac{14}{3}$

Final Answer:

$x = -\frac{14}{3}$

Would you like additional details on any part of this solution?

Here are five related questions for further exploration:

1. How would you solve this equation if there were parentheses around the entire right side?
2. What if the equation was $4(2x + 8) = 2x + k$? How does changing constants affect the solution?
3. How would you check your solution by substituting $x = -\frac{14}{3}$ back into the original equation?
4. What if the equation were $4((2x + 8) - 4) = 2x + 4$? How would you solve it then?
5. How would the solution change if the equation had a different coefficient, say $5(2x + 8) = 2x + 4$?

Tip: Always distribute and combine like terms carefully to simplify each step accurately.