To expand the expression \((x^2 + 3x + 1)(2x^4 + 2x + 2)\), we'll apply the distributive property (also known as the FOIL method for binomials, but this applies to larger polynomials as well). Here’s the step-by-step expansion:

\[
(x^2 + 3x + 1)(2x^4 + 2x + 2)
\]

Each term in the first polynomial \((x^2 + 3x + 1)\) must be multiplied by each term in the second polynomial \((2x^4 + 2x + 2)\).

1. Multiply \(x^2\) by each term in the second polynomial:
   \[
   x^2 \cdot (2x^4 + 2x + 2) = 2x^6 + 2x^3 + 2x^2
   \]

2. Multiply \(3x\) by each term in the second polynomial:
   \[
   3x \cdot (2x^4 + 2x + 2) = 6x^5 + 6x^2 + 6x
   \]

3. Multiply \(1\) by each term in the second polynomial:
   \[
   1 \cdot (2x^4 + 2x + 2) = 2x^4 + 2x + 2
   \]

Now, add all the terms together:

\[
2x^6 + 6x^5 + 2x^4 + 2x^3 + (2x^2 + 6x^2) + (6x + 2x) + 2
\]

Simplify:

\[
2x^6 + 6x^5 + 2x^4 + 2x^3 + 8x^2 + 8x + 2
\]

So, the expanded form is:

\[
2x^6 + 6x^5 + 2x^4 + 2x^3 + 8x^2 + 8x + 2
\]

Would you like more details on how each step was done, or do you have any questions?

Here are 5 related questions to explore further:
1. How do you factor polynomials like the one expanded here?
2. What is the degree of the polynomial after expansion?
3. How do you solve polynomials by factoring or using the quadratic formula?
4. What is the difference between expanding and factoring polynomials?
5. How does multiplying polynomials relate to solving equations?

**Tip:** When expanding polynomials, it's crucial to carefully track and combine like terms to avoid mistakes!

Explore the step-by-step process of expanding the polynomial (x^2 + 3x + 1)(2x^4 + 2x + 2) using the distributive property. This problem covers key algebraic concepts like polynomial multiplication and combining like terms. Suitable for students in Grades 10-12.

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