To factor the quadratic expression \(4x^2 + 28y + 49y^2\), we can start by observing that it resembles a perfect square trinomial. Here's a step-by-step approach to factor it:

### 1. Identify the terms in the trinomial
The given trinomial is:

\[
4x^2 + 28y + 49y^2
\]

This can be rewritten by recognizing the coefficients and terms:

- The first term is \(4x^2\), which can be written as \((2x)^2\).
- The last term is \(49y^2\), which can be written as \((7y)^2\).
- The middle term is \(28y\).

### 2. Check for a perfect square trinomial
For it to be a perfect square trinomial, the middle term should be \(2ab\), where \(a\) and \(b\) are the square roots of the first and last terms, respectively.

- Here, \(a = 2x\) and \(b = 7y\).
- Then \(2ab = 2(2x)(7y) = 28xy\).

Since the middle term matches \(28xy\), we confirm that the expression is a perfect square trinomial.

### 3. Factor the trinomial
The expression \(4x^2 + 28xy + 49y^2\) can be factored as:

\[
(2x + 7y)^2
\]

### Final Factored Form
\[
4x^2 + 28xy + 49y^2 = (2x + 7y)^2
\]

Would you like more details on this solution or have any other questions? Here are some related questions for further exploration:

1. How do you identify perfect square trinomials?
2. What is the process for factoring general quadratic expressions?
3. How do you factor trinomials that are not perfect squares?
4. Can you explain the concept of completing the square for solving quadratic equations?
5. How can you use the distributive property to verify the factorization?

**Tip:** When factoring quadratic expressions, always check if it can be rewritten as a perfect square trinomial before using other factoring methods!

Learn how to factor the quadratic expression 4x² + 28xy + 49y² as a perfect square trinomial with a detailed step-by-step solution. This problem covers key algebraic concepts, including perfect square trinomials and factoring techniques. Suitable for students in Grades 8-10.

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