Let's solve each equation step by step:

1) (x - 2)(x - 3) + (x - 2) - 1 = 0

First, let's simplify the equation:

(x - 2)(x - 3) + (x - 2) - 1 = 0
(x^2 - 5x + 6) + (x - 2) - 1 = 0
x^2 - 5x + 6 + x - 2 - 1 = 0
x^2 - 4x + 3 = 0

Now, let's factor the quadratic equation:

(x - 3)(x - 1) = 0

Setting each factor equal to zero:

x - 3 = 0 or x - 1 = 0

Solving for x:

x = 3 or x = 1

Therefore, the solutions to the equation are x = 3 and x = 1.

2) (x + 2)^2 - 2x(2x + 3) = (x + 1)^2

Let's simplify the equation:

(x + 2)^2 - 2x(2x + 3) = (x + 1)^2
(x^2 + 4x + 4) - 2x(2x + 3) = (x^2 + 2x + 1)

Expanding and simplifying:

x^2 + 4x + 4 - 4x^2 - 6x = x^2 + 2x + 1
-x^2 - 4x - 6x + 4 = x^2 + 2x + 1
-x^2 - 10x + 4 = x^2 + 2x + 1

Rearranging the equation:

2x^2 + 12x - 3 = 0

Now, let's solve the quadratic equation:

Using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a

a = 2, b = 12, c = -3

x = (-12 ± √(12^2 - 4(2)(-3))) / 2(2)
x = (-12 ± √(144 + 24)) / 4
x = (-12 ± √168) / 4
x = (-12 ± 2√42) / 4
x = -3 ± √42 / 2

Therefore, the solutions to the equation are x = (-3 + √42) / 2 and x = (-3 - √42) / 2.