Let's simplify the given expression step by step. The expression is:

$7 + \left( (-8)^2 + 7 \right) + \frac{-2}{4}$

Step 1: Simplify $(-8)^2$

$(-8)^2 = 64$

Step 2: Add 64 and 7 inside the parentheses

$64 + 7 = 71$

So, now the expression becomes: $7 + 71 + \frac{-2}{4}$

Step 3: Simplify $\frac{-2}{4}$

$\frac{-2}{4} = -0.5$

Step 4: Add all terms together

$7 + 71 - 0.5 = 77.5$

Final Answer:

$\boxed{77.5}$

Would you like further details on any of these steps or have any questions?

Here are some related questions you might find helpful:

1. How does squaring a negative number work?
2. Why does $\frac{-2}{4}$ simplify to $-0.5$?
3. What are the rules for order of operations in expressions like this?
4. How can parentheses affect the result in mathematical expressions?
5. How can you simplify similar expressions with fractions?

Tip: Always remember to follow the order of operations, commonly remembered as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).