To simplify the given expressions:

(x-2)(x+3)
Expand using FOIL (First, Outer, Inner, Last):
= xx + x3 - 2x - 23
= x^2 + 3x - 2x - 6
= x^2 + 1x - 6
= x^2 + x - 6

(x-1)^2
Expand the square:
= (x-1)(x-1)
= xx - x1 - 1*x + 1
= x^2 - x - x + 1
= x^2 - 2x + 1

Subtract (x-1)^2 from (x-2)(x+3):
(x^2 + x - 6) - (x^2 - 2x + 1)
= x^2 + x - 6 - x^2 + 2x - 1
= 3x - 5

4(a+5)^2
Expand using FOIL:
= 4(a^2 + 10a + 25)
= 4a^2 + 40a + 100

4(a^2 + 40a)
= 4a^2 + 160a

Subtract 4(a^2 + 40a) from 4(a+5)^2:
(4a^2 + 40a + 100) - (4a^2 + 160a)
= 4a^2 + 40a + 100 - 4a^2 - 160a
= -120a + 100

Therefore, the simplified expressions are:

(x-2)(x+3) = x^2 + x - 6(x-1)^2 = x^2 - 2x + 1(x-2)(x+3) - (x-1)^2 = 3x - 54(a+5)^2 - 4(a^2 + 40a) = -120a + 100