To simplify the expression 3x^3 - 12x + 4x^2 + 4xy + y^2, we can combine like terms.

The like terms are the terms that have the same variables and exponents.

The terms 3x^3 and -12x are already simplified and cannot be combined with any other terms.

The terms 4x^2 and 4xy can be combined because they both have x as a variable. To combine them, we add their coefficients: 4x^2 + 4xy = 4x^2 + 4xy.

The term y^2 cannot be combined with any other terms because it is the only term with y as a variable.

Therefore, the simplified expression is 3x^3 - 12x + 4x^2 + 4xy + y^2.