To find the product of \(x^2 + 4x + 1\), you would multiply the expression by itself, which means squaring the entire expression:

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

Now, expand the expression:

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

Step-by-step:

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

2. Multiply the second term \(4x\) by each term in the second polynomial:
   \[
   4x \cdot (x^2 + 4x + 1) = 4x^3 + 16x^2 + 4x
   \]

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

Now, combine all the terms:
\[
x^4 + 4x^3 + x^2 + 4x^3 + 16x^2 + 4x + x^2 + 4x + 1
\]

Simplify by combining like terms:
\[
x^4 + (4x^3 + 4x^3) + (x^2 + 16x^2 + x^2) + (4x + 4x) + 1
\]
\[
x^4 + 8x^3 + 18x^2 + 8x + 1
\]

Thus, the product of \( (x^2 + 4x + 1)^2 \) is:

\[
x^4 + 8x^3 + 18x^2 + 8x + 1
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you factor a polynomial like \(x^2 + 4x + 1\)?
2. What is the general method for expanding any binomial expression?
3. Can we use the binomial theorem to solve \( (x^2 + 4x + 1)^2 \)?
4. How do you find the derivative of the polynomial \(x^4 + 8x^3 + 18x^2 + 8x + 1\)?
5. What are the roots of the polynomial \(x^2 + 4x + 1\)?

**Tip:** When expanding polynomials, always group and simplify like terms carefully to avoid mistakes.

Learn how to find the product of (x^2 + 4x + 1) by squaring the expression and expanding it step-by-step. This detailed solution covers polynomial multiplication, simplifying like terms, and using the polynomial expansion theorem. Suitable for students in Grades 9-11.

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