To solve the equations \( 2x \sqrt{2x + 3} = 3x^2 + 6x \) and \( 12x \sqrt{2x + 3} = 3x^2 + 6x + 1 \), we will handle them one at a time.

### First Equation
1. **Equation:**
\[
2x \sqrt{2x + 3} = 3x^2 + 6x
\]

2. **Isolate the square root:**
\[
\sqrt{2x + 3} = \frac{3x^2 + 6x}{2x}
\]

Notice that for \( x \neq 0 \), we can simplify:
\[
\sqrt{2x + 3} = \frac{3x + 6}{2}
\]

3. **Square both sides to eliminate the square root:**
\[
2x + 3 = \left( \frac{3x + 6}{2} \right)^2
\]

4. **Expand the right side:**
\[
2x + 3 = \frac{(3x + 6)^2}{4} = \frac{9x^2 + 36x + 36}{4}
\]

5. **Multiply both sides by 4 to clear the fraction:**
\[
8x + 12 = 9x^2 + 36x + 36
\]

6. **Rearranging gives:**
\[
9x^2 + 36x - 8x + 36 - 12 = 0
\]
\[
9x^2 + 28x + 24 = 0
\]

7. **Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):**
- Where \( a = 9, b = 28, c = 24 \).

Calculate the discriminant:
\[
b^2 - 4ac = 28^2 - 4 \cdot 9 \cdot 24 = 784 - 864 = -80
\]

Since the discriminant is negative, there are no real solutions for the first equation.

### Second Equation
1. **Equation:**
\[
12x \sqrt{2x + 3} = 3x^2 + 6x + 1
\]

2. **Isolate the square root:**
\[
\sqrt{2x + 3} = \frac{3x^2 + 6x + 1}{12x}
\]

3. **Square both sides:**
\[
2x + 3 = \left( \frac{3x^2 + 6x + 1}{12x} \right)^2
\]

4. **Simplifying:**
\[
2x + 3 = \frac{(3x^2 + 6x + 1)^2}{144x^2}
\]

5. **Multiply both sides by \( 144x^2 \):**
\[
(2x + 3) \cdot 144x^2 = (3x^2 + 6x + 1)^2
\]

6. **Expand both sides. For the left side:**
\[
288x^3 + 432x^2
\]

For the right side, you would expand it:
\[
(3x^2 + 6x + 1)^2 = 9x^4 + 36x^3 + 36x^2 + 12x + 1
\]

7. **Set both equations equal:**
\[
288x^3 + 432x^2 = 9x^4 + 36x^3 + 36x^2 + 12x + 1
\]

8. **Rearranging gives:**
\[
9x^4 - 252x^3 - 396x^2 - 12x - 1 = 0
\]

At this point, the second equation also resembles a complex polynomial. Since solving such quartic equations analytically is difficult, typically numerical methods or software tools are utilized. With no real solutions from the first equation and a challenging second equation, if you still wish to pursue a solution, numerical techniques (e.g., Newton's method) could be employed on the second equation \( 9x^4 - 252x^3 - 396x^2 - 12x - 1 = 0 \).

Please let me know if you'd like to proceed further or explore different methods or approximations!