To expand the expression $(4x^2 + 4x + 1)^2$, we apply the formula for squaring a binomial, $(a + b)^2 = a^2 + 2ab + b^2$. In this case, we have a trinomial instead of a binomial, so we will use distribution (i.e., multiply the trinomial by itself).

$(4x^2 + 4x + 1)^2 = (4x^2 + 4x + 1)(4x^2 + 4x + 1)$

Step 1: Multiply each term of the first factor by each term of the second factor:

1. $4x^2 \cdot 4x^2 = 16x^4$
2. $4x^2 \cdot 4x = 16x^3$
3. $4x^2 \cdot 1 = 4x^2$
4. $4x \cdot 4x^2 = 16x^3$
5. $4x \cdot 4x = 16x^2$
6. $4x \cdot 1 = 4x$
7. $1 \cdot 4x^2 = 4x^2$
8. $1 \cdot 4x = 4x$
9. $1 \cdot 1 = 1$

Step 2: Combine like terms:

$16x^4 + 16x^3 + 4x^2 + 16x^3 + 16x^2 + 4x + 4x^2 + 4x + 1$

Now, group similar powers of $x$:

$16x^4 + (16x^3 + 16x^3) + (4x^2 + 16x^2 + 4x^2) + (4x + 4x) + 1$

Simplify:

$16x^4 + 32x^3 + 24x^2 + 8x + 1$

Thus, the expanded form is:

$(4x^2 + 4x + 1)^2 = 16x^4 + 32x^3 + 24x^2 + 8x + 1$

Would you like more details or have any questions? Here are 5 related problems for practice:

1. Expand $(3x^2 + 2x + 1)^2$.
2. Expand $(2x + 3)^2$.
3. Expand $(x^2 + 5x + 6)^2$.
4. Factor the expression $16x^4 + 32x^3 + 24x^2 + 8x + 1$.
5. Find the derivative of $16x^4 + 32x^3 + 24x^2 + 8x + 1$.

Tip: When multiplying polynomials, carefully organize and combine like terms to avoid mistakes.