Let's solve each equation step by step.

### D. \((4x + 3)^2 = (9x - 2)(x + 3)\)

1. Expand both sides:
\[
(4x + 3)^2 = 16x^2 + 24x + 9
\]
\[
(9x - 2)(x + 3) = 9x^2 + 27x - 2x - 6 = 9x^2 + 25x - 6
\]

2. Set the expanded forms equal to each other:
\[
16x^2 + 24x + 9 = 9x^2 + 25x - 6
\]

3. Move all terms to one side to set the equation to 0:
\[
16x^2 + 24x + 9 - 9x^2 - 25x + 6 = 0
\]
\[
7x^2 - x + 15 = 0
\]

4. Solve the quadratic equation \(7x^2 - x + 15 = 0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[
a = 7, \, b = -1, \, c = 15
\]
\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 7 \cdot 15}}{2 \cdot 7}
\]
\[
x = \frac{1 \pm \sqrt{1 - 420}}{14}
\]
\[
x = \frac{1 \pm \sqrt{-419}}{14}
\]

Since the discriminant \(\sqrt{-419}\) is negative, there are no real solutions.

### E. \((4x + 3)^2 = (8x - 2)(2x + 1)\)

1. Expand both sides:
\[
(4x + 3)^2 = 16x^2 + 24x + 9
\]
\[
(8x - 2)(2x + 1) = 16x^2 + 8x - 4x - 2 = 16x^2 + 4x - 2
\]

2. Set the expanded forms equal to each other:
\[
16x^2 + 24x + 9 = 16x^2 + 4x - 2
\]

3. Move all terms to one side to set the equation to 0:
\[
16x^2 + 24x + 9 - 16x^2 - 4x + 2 = 0
\]
\[
20x + 11 = 0
\]

4. Solve for \(x\):
\[
20x = -11
\]
\[
x = -\frac{11}{20}
\]

### F. \((3x - 1)^2 = (x - 4)(9x - 1)\)

1. Expand both sides:
\[
(3x - 1)^2 = 9x^2 - 6x + 1
\]
\[
(x - 4)(9x - 1) = 9x^2 - x - 36x + 4 = 9x^2 - 37x + 4
\]

2. Set the expanded forms equal to each other:
\[
9x^2 - 6x + 1 = 9x^2 - 37x + 4
\]

3. Move all terms to one side to set the equation to 0:
\[
9x^2 - 6x + 1 - 9x^2 + 37x - 4 = 0
\]
\[
31x - 3 = 0
\]

4. Solve for \(x\):
\[
31x = 3
\]
\[
x = \frac{3}{31}
\]

### G. \((5x - 1)^2 = (5x - 2)(5x + 2)\)

1. Expand both sides:
\[
(5x - 1)^2 = 25x^2 - 10x + 1
\]
\[
(5x - 2)(5x + 2) = 25x^2 - 4
\]

2. Set the expanded forms equal to each other:
\[
25x^2 - 10x + 1 = 25x^2 - 4
\]

3. Move all terms to one side to set the equation to 0:
\[
25x^2 - 10x + 1 - 25x^2 + 4 = 0
\]
\[
-10x + 5 = 0
\]

4. Solve for \(x\):
\[
-10x = -5
\]
\[
x = \frac{1}{2}
\]

### Summary of Solutions:
- D: No real solutions.
- E: \(x = -\frac{11}{20}\)
- F: \(x = \frac{3}{31}\)
- G: \(x = \frac{1}{2}\)