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The expression provided is \( 25x^2 + 40x + 16 \). To factor this, let's follow a structured approach:

### Step 1: Check for a Greatest Common Factor (GCF)
There is no common factor among all terms other than 1, so we don't factor out a GCF.

### Step 2: Recognize the structure
The expression is a perfect square trinomial. It takes the form \( a^2 + 2ab + b^2 \), which factors into \( (a + b)^2 \).

We can try to rewrite this in that form:
- \( 25x^2 \) is the square of \( 5x \),
- \( 16 \) is the square of 4,
- \( 40x \) is twice the product of \( 5x \) and 4.

Thus, this trinomial fits the pattern \( (5x + 4)^2 \).

### Step 3: Factor the expression
Hence, we factor it as:
\[
25x^2 + 40x + 16 = (5x + 4)(5x + 4) = (5x + 4)^2
\]

### Final Answer:
\[
25x^2 + 40x + 16 = (5x + 4)(5x + 4)
\]

Do you need further explanation, or is there a particular step you'd like more details on?

Here are 5 related questions:
1. What is the difference between factoring perfect square trinomials and other quadratic expressions?
2. How can you identify whether a quadratic expression is a perfect square trinomial?
3. What are the steps for factoring trinomials that are not perfect squares?
4. Why is factoring important in solving quadratic equations?
5. Can you apply factoring to solve \( 25x^2 + 40x + 16 = 0 \)?

**Tip**: Always check for a greatest common factor (GCF) first before factoring a quadratic expression. It simplifies the problem!

Learn how to factor the perfect square trinomial 25x^2 + 40x + 16. This detailed solution explains each step of factoring, including recognizing the trinomial form and applying the perfect square formula. Suitable for Grades 8-10.

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