Let's solve the given equation:

\[
(2x + 3)^2 - 6x - 9 = 0
\]

### Step 1: Expand \((2x + 3)^2\)
Using the formula \((a + b)^2 = a^2 + 2ab + b^2\), we get:

\[
(2x + 3)^2 = (2x)^2 + 2(2x)(3) + 3^2 = 4x^2 + 12x + 9
\]

So the equation becomes:

\[
4x^2 + 12x + 9 - 6x - 9 = 0
\]

### Step 2: Simplify the equation
Combine like terms:

\[
4x^2 + (12x - 6x) + (9 - 9) = 0
\]

\[
4x^2 + 6x = 0
\]

### Step 3: Factor the equation
Factor out the common term \(2x\):

\[
2x(2x + 3) = 0
\]

### Step 4: Solve for \(x\)
Set each factor to zero:

1. \(2x = 0 \Rightarrow x = 0\)
2. \(2x + 3 = 0 \Rightarrow 2x = -3 \Rightarrow x = -\frac{3}{2}\)

### Final Answer:
The solutions to the equation are:

\[
x = 0 \quad \text{or} \quad x = -\frac{3}{2}
\]

Would you like more details on any of the steps?

Here are 5 related questions to explore further:
1. How do you solve quadratic equations by factoring?
2. What are the properties of perfect square trinomials?
3. How does the distributive property work in expanding expressions?
4. What is the quadratic formula, and how can it be applied?
5. How do you solve quadratic equations using the method of completing the square?

**Tip:** Always check your final answers by substituting them back into the original equation to ensure they satisfy it!

Explore the step-by-step solution to the quadratic equation (2x + 3)^2 - 6x - 9 = 0. Learn how to expand expressions, simplify equations, and use factoring to find the roots. Suitable for students in Grades 8-10.

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