Let's solve each of the given equations step-by-step.

### Equation A: \((x-5)^2 = (x+1)(x-2)\)

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

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

3. Subtract \(x^2\) from both sides:
\[
-10x + 25 = -x - 2
\]

4. Add \(10x\) to both sides:
\[
25 = 9x - 2
\]

5. Add 2 to both sides:
\[
27 = 9x
\]

6. Divide by 9:
\[
x = 3
\]

So, the solution to Equation A is \(x = 3\).

### Equation B: \((x+3)^2 = (x-1)(x+4)\)

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

2. Set the expanded forms equal to each other:
\[
x^2 + 6x + 9 = x^2 + 3x - 4
\]

3. Subtract \(x^2\) from both sides:
\[
6x + 9 = 3x - 4
\]

4. Subtract \(3x\) from both sides:
\[
3x + 9 = -4
\]

5. Subtract 9 from both sides:
\[
3x = -13
\]

6. Divide by 3:
\[
x = -\frac{13}{3}
\]

So, the solution to Equation B is \(x = -\frac{13}{3}\).

### Equation C: \((2x + 3)^2 = (4x + 1)(x - 2)\)

1. Expand both sides:
\[
(2x + 3)^2 = 4x^2 + 12x + 9
\]
\[
(4x + 1)(x - 2) = 4x^2 - 8x + x - 2 = 4x^2 - 7x - 2
\]

2. Set the expanded forms equal to each other:
\[
4x^2 + 12x + 9 = 4x^2 - 7x - 2
\]

3. Subtract \(4x^2\) from both sides:
\[
12x + 9 = -7x - 2
\]

4. Add \(7x\) to both sides:
\[
19x + 9 = -2
\]

5. Subtract 9 from both sides:
\[
19x = -11
\]

6. Divide by 19:
\[
x = -\frac{11}{19}
\]

So, the solution to Equation C is \(x = -\frac{11}{19}\).

In summary:
- Solution to Equation A: \(x = 3\)
- Solution to Equation B: \(x = -\frac{13}{3}\)
- Solution to Equation C: \(x = -\frac{11}{19}\)