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We are tasked with expanding the expression \((5z^2 + 3z + 2)^2\).

Let's break it down step by step:

The given expression is a binomial square:

\[
(5z^2 + 3z + 2)^2
\]

We can expand this by applying the formula \((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc\), where \(a = 5z^2\), \(b = 3z\), and \(c = 2\). Let's expand it now:

### Step 1: Square each term
1. \((5z^2)^2 = 25z^4\)
2. \((3z)^2 = 9z^2\)
3. \(2^2 = 4\)

### Step 2: Multiply each pair of different terms and double them
1. \(2 \times (5z^2) \times (3z) = 30z^3\)
2. \(2 \times (5z^2) \times 2 = 20z^2\)
3. \(2 \times (3z) \times 2 = 12z\)

### Step 3: Combine all terms
\[
25z^4 + 30z^3 + (9z^2 + 20z^2) + 12z + 4
\]

This simplifies to:

\[
25z^4 + 30z^3 + 29z^2 + 12z + 4
\]

Thus, the correct expanded form is:

\[
25z^4 + 30z^3 + 29z^2 + 12z + 4
\]

The correct answer from the given choices is the last option.

Would you like further details or explanations? Here are some related questions:

1. How do you apply the binomial square formula to more complex expressions?
2. What is the general formula for expanding polynomials?
3. Could the distributive property be used to expand \((5z^2 + 3z + 2)^2\) differently?
4. How would the result change if the powers of \(z\) were higher (e.g., \(5z^3 + 3z^2 + 2\))?
5. How can you simplify polynomial expressions after expansion?

**Tip:** When expanding binomials, remember to handle each term's powers carefully and combine like terms at the end to simplify the expression.

What expression is equivalent to (5z^2 + 3z + 2)^2?

Learn how to expand (5z^2 + 3z + 2)^2 step-by-step using the binomial expansion method. This problem covers key algebraic concepts, including polynomial expansion and the binomial square formula. Suitable for students in Grades 9-12.

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