To solve the expression

$\left(\sqrt{5} + 4\right)^2 + \left(\sqrt{5} - 4\right)^2$

we'll expand each squared term individually and then add the results.

Step 1: Expanding $\left(\sqrt{5} + 4\right)^2$

Using the formula $(a + b)^2 = a^2 + 2ab + b^2$:

$\left(\sqrt{5} + 4\right)^2 = \left(\sqrt{5}\right)^2 + 2 \cdot \sqrt{5} \cdot 4 + 4^2$ $= 5 + 8\sqrt{5} + 16$ $= 21 + 8\sqrt{5}$

Step 2: Expanding $\left(\sqrt{5} - 4\right)^2$

Similarly, using $(a - b)^2 = a^2 - 2ab + b^2$:

$\left(\sqrt{5} - 4\right)^2 = \left(\sqrt{5}\right)^2 - 2 \cdot \sqrt{5} \cdot 4 + 4^2$ $= 5 - 8\sqrt{5} + 16$ $= 21 - 8\sqrt{5}$

Step 3: Adding the Results

Now, we add the two expanded expressions:

$\left(\sqrt{5} + 4\right)^2 + \left(\sqrt{5} - 4\right)^2 = (21 + 8\sqrt{5}) + (21 - 8\sqrt{5})$

Notice that the $8\sqrt{5}$ terms cancel each other out:

$= 21 + 21 = 42$

Final Answer

$\left(\sqrt{5} + 4\right)^2 + \left(\sqrt{5} - 4\right)^2 = 42$

Would you like more details, or do you have any questions?

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Here are some related questions to consider:

1. How does the distributive property work in simplifying expressions like this?
2. What other expressions simplify due to symmetric terms canceling each other?
3. How can the concept of conjugates simplify expressions with radicals?
4. What is the importance of identifying terms that cancel in algebraic expressions?
5. How would this approach change if we had different constants instead of $4$ in the expression?

Tip: Recognizing conjugate pairs in expressions with square roots can often simplify the process by eliminating terms involving the square roots.