https://file.aiquickdraw.com/mathbot/file/172736468284424k1ccwf.jpg

The equation given is:

\[
(2x + 5)^2 = (2x + 3)(2x - 1)
\]

### Step 1: Expand both sides.

**Left-hand side:**
\[
(2x + 5)^2 = (2x + 5)(2x + 5) = 4x^2 + 10x + 10x + 25 = 4x^2 + 20x + 25
\]

**Right-hand side:**
\[
(2x + 3)(2x - 1) = 2x(2x) + 2x(-1) + 3(2x) + 3(-1)
\]
\[
= 4x^2 - 2x + 6x - 3 = 4x^2 + 4x - 3
\]

### Step 2: Set up the equation.
After expanding both sides, the equation becomes:
\[
4x^2 + 20x + 25 = 4x^2 + 4x - 3
\]

### Step 3: Simplify.
Subtract \(4x^2\) from both sides:
\[
20x + 25 = 4x - 3
\]

### Step 4: Move all terms involving \(x\) to one side and constant terms to the other:
\[
20x - 4x = -3 - 25
\]
\[
16x = -28
\]

### Step 5: Solve for \(x\):
\[
x = \frac{-28}{16} = \frac{-7}{4}
\]

### Final Answer:
\[
x = -\frac{7}{4}
\]

Would you like a more detailed breakdown of any step? Let me know if you have further questions!

Here are five questions related to this topic:
1. How do you expand binomials using the distributive property?
2. What are common methods to solve quadratic equations?
3. How can we verify if a solution to an equation is correct?
4. What is the difference between factoring and expanding expressions?
5. How do you solve equations with higher powers of \(x\)?

**Tip:** Always check your solution by substituting the value of \(x\) back into the original equation to ensure it satisfies both sides.

Step-by-step guide to solving the quadratic equation (2x + 5)^2 = (2x - 3)(2x - 1). Expand both sides, simplify, and solve for x. Learn about expanding binomials and applying algebraic techniques suitable for students in Grades 8-10.

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