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.

Learn how to expand the expression (4x^2 + 4x + 1)^2 with a detailed step-by-step solution. This problem covers key algebraic concepts, including polynomial expansion and using the distributive property. Suitable for students in Grades 9-11.

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