Yes, those are the three things you need to do.

Now, the first set, (a), is not a vector space; it is missing the zero function.

For (b), can you add two polynomials of degree less than 3 to get a polynomial of degree 3 or more? Does multiplication by a non-zero scalar change the degree of a polynomial?

(c), again, is a vector space. You just need to do the checks and think about why they work.

(d) is not. I will leave you to work out why...