U is the subspace spanned by , , and . If those three vectors are independent then they form a basis for U themselves.
Suppose [tex]a(x^3+ 4x^2- x+ 3)+ b(x^3+ 5x^2+ 5)+ c(x^3+ 10x^2+ 5)= 0[/itex]. That gives . Since , , , and 1 are independent themselves, we must have a+ b+ c= 0, 4a+ 6b+ 10c= 0, -a= 0, and 3a+ 5b+ 5c= 0. From the third equation, a= 0. so the others are b+ c= 0, 6b+ 10c= 0, and 5b+ 5c= 5(b+ c)= 0. The first and last equations are satisfied as long as c= -b but that makes the other equation 6b+ 10(-b)= -4b= 0 so b= 0 and c= 0. Yes, those three vectors are independent and so form a basis for U which is three dimensional.
Do the same thing with V.