Question

How do you find an equation of a polynomial of degree 4 that has 1) no zeros, 2) one zero, and the like?

Answer 1

If it has no zeros, it means it has no zeros in reals, but it has it four zeros in complex numbers.
So you only need to multiply two polynomys of degree 4 and no zeros:
For example:
(x^2+1)(x^2+1)=x^4+2x^2+1
It has no zeros
You can't find a polynomial of degree 4 with 1 zero in real, but you can find one with two equals zeros.
for example, x^2+2x+1=(x+1)(x+1) has two zeros at x=-1
Then
(x^2+1)(x+1)^2=(x^2+1)(x^2+2x+1)=x^4+2x^3+2x^2+2x+1