Let's solve the equation:

\[
|-4a - 4| - 4 = 32
\]

### Step 1: Isolate the absolute value term
Add 4 to both sides to eliminate the -4 on the left side:

\[
|-4a - 4| = 32 + 4
\]
\[
|-4a - 4| = 36
\]

### Step 2: Solve the absolute value equation
We know that if \( |x| = y \), then \( x = y \) or \( x = -y \). So we need to split this into two cases:

1. \( -4a - 4 = 36 \)
2. \( -4a - 4 = -36 \)

### Case 1: \( -4a - 4 = 36 \)

Solve for \( a \):

\[
-4a = 36 + 4
\]
\[
-4a = 40
\]
\[
a = \frac{40}{-4}
\]
\[
a = -10
\]

### Case 2: \( -4a - 4 = -36 \)

Solve for \( a \):

\[
-4a = -36 + 4
\]
\[
-4a = -32
\]
\[
a = \frac{-32}{-4}
\]
\[
a = 8
\]

### Final Solution:
The two possible solutions are:
\[
a = -10 \quad \text{or} \quad a = 8
\]

Would you like further details on any of these steps?

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Here are 5 questions related to this problem:
1. How do you solve equations involving absolute values?
2. What happens if the absolute value of an expression equals a negative number?
3. Can you explain the steps for solving linear equations?
4. How do absolute value properties change when dealing with inequalities?
5. Can absolute value functions have more than two solutions?

**Tip:** Always check both solutions in the original equation, especially for absolute value problems.

Discover how to solve the absolute value equation |-4a - 4| - 4 = 32 with a detailed step-by-step breakdown. This guide covers algebraic concepts like isolating the absolute value, solving for multiple cases, and checking solutions. Ideal for students in Grades 8-10.

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