We want to express \((4 + 4x)^2\) in the form \( a(b + cx)^2 \), where \(a\), \(b\), and \(c\) are constants.

Starting with the expression:

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

Factor out the greatest common factor (GCF) from \(4 + 4x\), which is 4:

\[
4(1 + x)
\]

Now square the entire expression:

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

So, \((4 + 4x)^2\) can be rewritten as:

\[
16(1 + x)^2
\]

Thus, the expression in the form \( a(b + cx)^2 \) is:

\[
a = 16, \quad b = 1, \quad c = 1
\]

### Final Answer:
\[
(4 + 4x)^2 = 16(1 + x)^2
\]

Would you like further details or have any questions?

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Here are some related questions you might explore:

1. How do you expand \((1 + x)^2\)?
2. What happens when you apply the binomial theorem to \((4 + 4x)^2\)?
3. How do you find the derivative of \((4 + 4x)^2\)?
4. What is the factorization of \((4 + 4x)^2\)?
5. How do you solve \((4 + 4x)^2 = 0\)?

**Tip:** When factoring, always look for the greatest common factor (GCF) first to simplify the expression.

Learn how to express (4 + 4x)^2 in the form a(b + cx)^2 using algebraic manipulation and the binomial theorem. This problem covers key algebraic concepts, including the binomial expansion formula and quadratic equations. Suitable for students in Grades 9-12.

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