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The expression given is \((8x^4 + 1)^2\). To expand this, we will use the algebraic identity for the square of a binomial:

\[
(a + b)^2 = a^2 + 2ab + b^2
\]

In this case, \(a = 8x^4\) and \(b = 1\).

Let's expand it step by step:

\[
(8x^4 + 1)^2 = (8x^4)^2 + 2 \cdot (8x^4) \cdot 1 + 1^2
\]

1. First, calculate \((8x^4)^2\):
   \[
   (8x^4)^2 = 64x^8
   \]

2. Next, calculate \(2 \cdot (8x^4) \cdot 1\):
   \[
   2 \cdot 8x^4 \cdot 1 = 16x^4
   \]

3. Finally, calculate \(1^2\):
   \[
   1^2 = 1
   \]

Now, combine all the terms:

\[
(8x^4 + 1)^2 = 64x^8 + 16x^4 + 1
\]

So the expanded form of the expression is:

\[
64x^8 + 16x^4 + 1
\]

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How would you expand \((2x^2 + 3)^2\)?
2. What is the general formula for expanding \((ax^n + b)^2\)?
3. How does this expansion relate to the binomial theorem?
4. Can you factorize \(64x^8 + 16x^4 + 1\) further?
5. What happens if you multiply \((8x^4 + 1)\) by another polynomial?

**Tip:** When expanding binomials, ensure that you correctly apply the identity \((a + b)^2 = a^2 + 2ab + b^2\), as this helps in avoiding common mistakes.

Explore how to expand the expression (8x^4 + 1)^2 using the square of a binomial theorem with a detailed step-by-step solution. This problem covers algebraic concepts including the binomial theorem and is suitable for Grades 9-12.

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