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Solve this using factoring perfect trinomial 

The expression given is:

\[
15x^2 + 30x
\]

To use the method of factoring as a perfect trinomial, we need to ensure the expression fits the form of a perfect square trinomial, which is generally:

\[
a^2 + 2ab + b^2 = (a + b)^2
\]

However, the expression \(15x^2 + 30x\) is a binomial, not a trinomial, and it lacks a constant term that could complete the square. To factor it as a perfect square trinomial, you would first need to transform it into a trinomial by adding and subtracting a constant term.

### Step 1: Factor out the GCF
First, factor out the greatest common factor (GCF), which is \(15x\):

\[
15x(x + 2)
\]

### Step 2: Complete the square
To create a perfect square trinomial inside the parentheses, recognize that the term inside the parentheses is \(x + 2\).

To complete the square, add and subtract \(1\) inside the parentheses:

\[
15x \left( x^2 + 2x + 1 - 1 \right)
\]

This simplifies to:

\[
15x \left( (x + 1)^2 - 1 \right)
\]

So, the expression doesn't naturally form a perfect square trinomial without modifying the original terms. As it stands, it cannot be factored as a perfect trinomial because it's not in the correct form.

However, using the method of completing the square or by adding a specific term, you can transform and factor the expression in a form close to a perfect square.

Would you like more details on completing the square or any other related process?

Here are some related questions to deepen your understanding:

1. How can you identify a perfect square trinomial?
2. What are the steps to complete the square for the expression \(x^2 + 6x\)?
3. Can the expression \(15x^2 + 30x + 25\) be factored as a perfect square trinomial?
4. What role does the constant term play in forming a perfect square trinomial?
5. How would you solve the equation \(15x^2 + 30x = 0\) using factoring?

**Tip:** For an expression to be factored as a perfect square trinomial, it needs to have three terms where the first and last terms are perfect squares, and the middle term is twice the product of their square roots.

Explore how to factor perfect trinomials like 15x^2 + 30x. Understand the method of completing the square and transforming binomials into trinomials. Suitable for high school students studying algebra.

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