Determine the basis of a subset V in a vector space of R3

Question

I have to determine a basis of a given polinomyal subset. V is defined as:

link

I don't know how to start, maybe it could be helpfull to know that V is the subset of polynomial in the following form:

(x+1)g(x) with g(x) polynomial with degree at most 2

Answer 1

$$P (x)=ax^3+bx^2+cx+d $$ is an element of $V $ if only if

$$P (-1)=-a+b-c+d=0$$ thus $$d=a+c-b $$ and then $$P (x)=a (x^3+1)+b (x^2-1)+c (x+1) $$

$$\implies V=span (x^3+1,x^2-1,x+1) $$

theses polynoms are independant and forms a base of $V $.

Answer 2

$e_1 = x+1\\ e_2 = (x+1)x = x^2 + x$

etc.