The problem asks to solve the following equation for $x$:

$\frac{-3x - 4}{4} = 2 + \frac{x}{4} + \frac{3x + 4}{4}.$

Step 1: Combine like terms on the right-hand side

First, write all terms over a common denominator (4) to combine:

$\text{Right-hand side: } 2 + \frac{x}{4} + \frac{3x + 4}{4} = \frac{8}{4} + \frac{x}{4} + \frac{3x + 4}{4}.$

Combine all terms into a single fraction: $\frac{8}{4} + \frac{x}{4} + \frac{3x + 4}{4} = \frac{8 + x + 3x + 4}{4} = \frac{12 + 4x}{4}.$

Thus, the equation becomes: $\frac{-3x - 4}{4} = \frac{12 + 4x}{4}.$

Step 2: Eliminate the denominator

Multiply through by 4 to simplify: $-3x - 4 = 12 + 4x.$

Step 3: Isolate $x$

Bring all terms involving $x$ to one side and constants to the other: $-3x - 4x = 12 + 4,$ $-7x = 16.$

Solve for $x$: $x = -\frac{16}{7}.$

Final Answer:

$x = -\frac{16}{7}.$

Would you like me to explain any step further or verify the solution?

Here are 5 related questions to explore:

1. How do you verify the solution of an equation involving fractions?
2. What methods can simplify equations with multiple terms in the denominator?
3. How do you convert improper fractions to mixed numbers (e.g., $-\frac{16}{7}$)?
4. What happens if you introduce decimals to solve such an equation?
5. How would the process change if the fractions had different denominators?

Tip: Always combine terms into a single fraction when dealing with equations involving fractions—it makes simplifying much easier.