Let pn denote the set of all polynomials of degree at most n with real coefficients (i. e. polynomials of the form a0+a1t+a2t 2++ant n for a0, a1, an r) and let s = {t 2 , 1+t, t3+4}. a. is the set s linearly independent in p3?

We can consider each unique as the as the -th unit vector. So your set can be considered as the vectors 



Then check for independence your favorite way. In this case, I'll see if the linear map A of the new basis vectors doesn't map to a subspace via the determinant not being zero:



So they are linear independent. 

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