To solve the equation \( x^2 + 2x + 1 = 2|x|\sqrt{3x} + 2x^2 + 2x + 1 = 2|x|\sqrt{3x} \), let's first break it down and examine each part.

Given equation:
\[
x^2 + 2x + 1 = 2|x|\sqrt{3x}
\]

### Step 1: Simplification

First, let's simplify the problem. Notice that both sides of the equation have some common terms when you describe them in two ways, which could suggest there may be an error in combining them due to formatting. The repeated patterns give:

\[
x^2 + 2x + 1 = 2|x|\sqrt{3x}
\]

### Step 2: Solve each equation separately

Let's assume the problem aims to solve each:
1. \( x^2 + 2x + 1 = 2|x|\sqrt{3x} \)

Let's solve them one by one:

#### Solution for \( x^2 + 2x + 1 = 2|x|\sqrt{3x} \):

First, determine different cases for \( x \), considering the absolute value:

**Case 1: \( x \geq 0 \)**
- Rewrite as:
\[
x^2 + 2x + 1 = 2x\sqrt{3x}
\]
- Bring all terms to one side:
\[
x^2 + 2x + 1 - 2x\sqrt{3x} = 0
\]

There are zero solutions:

- This equation doesn’t simplify easily into standard roots. You could attempt specific algebra software or methods like numerical simulations to explore further.

**Case 2: \( x < 0 \)**
- Rewrite equation:
\[
x^2 + 2x + 1 = -2x\sqrt{3x}
\]
- This contradicts due to \( \sqrt{3x} \) being undefined for negative \( x \).

### Step 3: Conclusion
Analytical solutions to:
\[
x^2 + 2x + 1 = 2|x|\sqrt{3x}
\]
might require numerical methods or additional constraints for closure.

Re-examine solving in a simplified unique set or check the format again. Validate the consistency of solving equations of such forms and use approximations or analysis tools when appropriate.