You need to verify that it is
1: Show that
2: that it is closed
You are correct that i) is a subspace. The equation of the plane given shows you what properties the vector must satisfy.
This "system" of equations has 4 variables and 1 equation so it will have 3 free variables. Let be the free variables then
Now writing the vector solution gives
As you can check these vectors are the basis of you space. Each on by itself satisfy the equation above.
For ii) it is not a subspace as it is not linear. Note that
Now show that
For the last one here is a hint:
This gives you a form for the polynomial in your space.