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Let S be the subspace of ℝ4 consisting of the solutions to the following system of...

Let S be the subspace of ℝ4 consisting of the solutions to the following system of equations:

Given the following polynomials p(x) and q(x), find a polynomial r(x) in ℙ3 so that {p(x), q(x), r(x)} is linearly independent and a non-zero polynomial s(x) in ℙ3 so that {p(x), q(x), s(x)} is linearly dependent:

Use the '^' character to indicate an exponent, e.g. 5x^2−2x+1.
p(x)=--4x^3-2x^2+10x+3
q(x)=4x^3-10x+10


Give a basis for S.

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