To simplify the given expressions, let's break them down step by step:

1) x² * 2y:
When multiplying two terms, we multiply the coefficients (numbers) and add the exponents of the variables. In this case, the coefficient of x² is 1, and the coefficient of y is 2. The exponent of x is 2, and the exponent of y is 1. Therefore, we have:
x² * 2y = 1 * 2 * x² * y = 2x²y

2) -2y * 2y:
Again, we multiply the coefficients and add the exponents. The coefficient of -2y is -2, and the exponent of y is 1. Therefore, we have:
-2y * 2y = -2 * 2 * y * y = -4y²

3) -2y * x²:
Similarly, we multiply the coefficients and add the exponents. The coefficient of -2y is -2, and the exponent of x is 2. Therefore, we have:
-2y * x² = -2 * x² * y = -2x²y

So, the simplified expressions are:
x² * 2y = 2x²y
-2y * 2y = -4y²
-2y * x² = -2x²y