To simplify the expressions, we can expand and combine like terms.

For the first expression, 2x(x - 2) - (x - 2)^2, we can start by expanding (x - 2)^2:
(x - 2)^2 = (x - 2)(x - 2) = x(x - 2) - 2(x - 2) = x^2 - 2x - 2x + 4 = x^2 - 4x + 4

Now we substitute this back into the original expression:
2x(x - 2) - (x - 2)^2 = 2x(x - 2) - (x^2 - 4x + 4)

Expanding 2x(x - 2):
2x(x - 2) = 2x^2 - 4x

Substituting this back into the expression:
2x(x - 2) - (x^2 - 4x + 4) = 2x^2 - 4x - (x^2 - 4x + 4)

Now we can combine like terms:
2x^2 - 4x - x^2 + 4x - 4 = x^2 - 4

So the simplified expression is x^2 - 4.

For the second expression, 3x(x + 2) - 5(x + 2)^2, we can start by expanding (x + 2)^2:
(x + 2)^2 = (x + 2)(x + 2) = x(x + 2) + 2(x + 2) = x^2 + 2x + 2x + 4 = x^2 + 4x + 4

Now we substitute this back into the original expression:
3x(x + 2) - 5(x + 2)^2 = 3x(x + 2) - 5(x^2 + 4x + 4)

Expanding 3x(x + 2):
3x(x + 2) = 3x^2 + 6x

Substituting this back into the expression:
3x(x + 2) - 5(x^2 + 4x + 4) = 3x^2 + 6x - 5(x^2 + 4x + 4)

Now we can distribute the -5:
3x^2 + 6x - 5x^2 - 20x - 20

Combining like terms:
(3x^2 - 5x^2) + (6x - 20x) - 20 = -2x^2 - 14x - 20

So the simplified expression is -2x^2 - 14x - 20.