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Let's simplify and calculate the value of the given expression step-by-step. The expression is:

\[
\frac{(\sqrt{3} - 4)(\sqrt{3} + 4) + \left(2^3 \sqrt{2}\right)^2}{-2}
\]

### Step 1: Simplify \((\sqrt{3} - 4)(\sqrt{3} + 4)\)

This part can be simplified using the difference of squares formula:
\[
(a - b)(a + b) = a^2 - b^2
\]
In this case, \(a = \sqrt{3}\) and \(b = 4\). So:
\[
(\sqrt{3})^2 - 4^2 = 3 - 16 = -13
\]

### Step 2: Simplify \((2^3 \sqrt{2})^2\)

First, calculate \(2^3\):
\[
2^3 = 8
\]
So, \(2^3 \sqrt{2} = 8\sqrt{2}\).

Now, square this term:
\[
(8\sqrt{2})^2 = 8^2 \cdot (\sqrt{2})^2 = 64 \cdot 2 = 128
\]

### Step 3: Substitute and Simplify the Expression

Now we can substitute back into the original expression:
\[
\frac{-13 + 128}{-2} = \frac{115}{-2} = -57.5
\]

### Final Answer
\[
-57.5
\]

Would you like a more detailed breakdown of each step or have any questions?

Here are 5 related questions for further practice:

1. How does the difference of squares formula apply to real-life problems?
2. What are some examples of simplifying square roots in algebraic expressions?
3. How do powers affect square root terms in expressions?
4. Can you simplify expressions involving complex numbers using similar steps?
5. How would this calculation change if there were cube roots instead of square roots?

**Tip:** Always simplify each part of a complex expression individually before combining the results; it minimizes errors and clarifies each step.

uprość i oblicz wartość wyrażenia: \(\frac{(\sqrt{3} - 4)(\sqrt{3} + 4) + \left(2^3 \sqrt{2}\right)^2}{-2}\)

Learn how to simplify and calculate the expression \(\frac{(\sqrt{3} - 4)(\sqrt{3} + 4) + \left(2^3 \sqrt{2}\right)^2}{-2}\) with a step-by-step solution. This problem involves algebraic concepts, including the difference of squares and exponentiation. Suitable for students in Grades 10-12.

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