The equation in the image is:

$|x^2 - 2x - 4| = 4$

Step-by-Step Solution:

1. Understanding Absolute Value: The equation $|A| = B$ implies two cases:

   • $A = B$
   • $A = -B$

   Here, $A = x^2 - 2x - 4$ and $B = 4$.

2. Case 1: $x^2 - 2x - 4 = 4$: Simplify: $x^2 - 2x - 8 = 0$

   Factorize: $(x - 4)(x + 2) = 0$

   Solutions: $x = 4 \quad \text{or} \quad x = -2$

3. Case 2: $x^2 - 2x - 4 = -4$: Simplify: $x^2 - 2x = 0$

   Factorize: $x(x - 2) = 0$

   Solutions: $x = 0 \quad \text{or} \quad x = 2$

4. Combine Solutions: The solutions are: $x = -2, 0, 2, 4$

Final Answer:

The solutions are: $x = -2, 0, 2, 4$

Let me know if you have questions or need further clarification.

Related Questions:

1. How do we solve equations involving absolute values in general?
2. What is the significance of factoring in quadratic equations?
3. Could we use the quadratic formula instead of factoring?
4. How do we verify the solutions in absolute value equations?
5. What is the graphical interpretation of this solution?

Tip:

Always check your solutions when solving absolute value equations to ensure they satisfy the original equation.