Answers:

1. Yes.

4xy² + 12y can be factored into 2(2xy² + 6y). Since x and y are integers, and the products and sums of integers are integers, 2xy² + 6y is also an integer. According to the definition of even integer, an integer is even iff that integer is equal to 2k for some integer k.

 4xy² + 12y = 2(2xy² + 6y),

and 2xy² + 6y is an integer; therefore, 4xy² + 12y is even.

2. Yes.

As n² is even, n must be even. Therefore, n = 2k for some integer k from the definition.
Since then we will have n² = (2k)² = 4k², 4k² is a multiple of 4, which is divisible by 4.

3. The first five prime numbers are 2, 3, 5, 7, 11.
    The first five composite numbers are 4, 6, 8, 9, 10.