Let's solve the given expression step by step:

The expression is:

$|2 - 5| - \left( \frac{6}{2} + 1 \right)^2$

Step 1: Simplify the absolute value part

First, calculate the absolute value:

$|2 - 5| = |-3| = 3$

So, the expression becomes:

$3 - \left( \frac{6}{2} + 1 \right)^2$

Step 2: Simplify inside the parentheses

Next, simplify $\frac{6}{2} + 1$:

$\frac{6}{2} = 3$

$3 + 1 = 4$

So now, the expression is:

$3 - 4^2$

Step 3: Square the 4

$4^2 = 16$

Step 4: Final calculation

Now subtract:

$3 - 16 = -13$

Final Answer:

$\boxed{-13}$

Would you like further clarification on any step? Here are some related problems for practice:

1. $|7 - 12| - \left(\frac{8}{4} + 2\right)^2$
2. $|4 - 9| - \left(\frac{10}{5} + 3\right)^2$
3. $|5 - 3| - \left(\frac{9}{3} + 4\right)^2$
4. $|10 - 15| - \left(\frac{12}{6} + 5\right)^2$
5. $|8 - 2| - \left(\frac{14}{7} + 6\right)^2$

Tip: Always solve operations inside parentheses and apply exponents before handling addition or subtraction outside.