Let v be the vector space p3[x] of polynomials in x with degree less than 3 and w be the subspace w=span{1−5x−4x^2,−(1+x+5x^2)}. find a polynomial q(x) in v∖w. q(x)=

By definition, the span of two vectors is the set of all possible linear combinations of said vectors. So, any element in W is obtaining by choosing two numbers a,b and building

(a)

So, you can show a nonzero polynomial in W by choosing any values of a and b, as long as they're not both zero. To keep it as simple as possible, we can choose for example a=1, b=0 and we obtain one of the base vectors of W:

(b)

Factoring the powers of x, we see that a generic polynomial in W looks like

So, we must build a polynomial

such that it is not possible that

For example, let's try the polynomial

We should solve the system

And you can check that it has no solutions. So, the polynomial does not belong to W. On the other hand, it surely belongs to V, because it is a polynomial with degree less than 3. So, the polynomial belongs to V/W.