The given operation $a * b$ is defined as:

$a * b = (a + b)^2 - (b + a^2)$

We are asked to compute the value of $(1 * 2) * 3$. Let's first break it down.

Step 1: Compute $1 * 2$

Substitute $a = 1$ and $b = 2$ into the formula for $a * b$:

$1 * 2 = (1 + 2)^2 - (2 + 1^2)$ $1 * 2 = 3^2 - (2 + 1)$ $1 * 2 = 9 - 3 = 6$

Step 2: Compute $(1 * 2) * 3$

Now, we substitute $a = 6$ and $b = 3$ into the formula again:

$6 * 3 = (6 + 3)^2 - (3 + 6^2)$ $6 * 3 = 9^2 - (3 + 36)$ $6 * 3 = 81 - 39 = 42$

Thus, the value of $(1 * 2) * 3$ is 42.

Would you like more details on any part of the solution?

Here are 5 related questions to explore further:

1. How would the result change if the operation was defined differently?
2. Can you verify the operation for different values of $a$ and $b$?
3. What are the properties of this custom operation $*$? Is it commutative or associative?
4. How would you simplify the expression if the powers were different?
5. How does this operation compare to standard arithmetic operations?

Tip: Always substitute and simplify step by step to avoid errors when dealing with custom operations.