Find a basis for the space $\displaystyle \{(a+c, b+d, a+d, b+c)\vert a,b,c,d \epsilon \mathbb{R}\}$

Again, literally no clue on this; my lecturer really can't teach very well. Also, might as well put the other problem I have in here; it's in the same vein (I think!)

Find a basis for the space of all polynomials of degree at most 5 for which $\displaystyle f(0)=f(1)=f(-1)$.