You need to find the coefficents c's first. As in
ClearAll[c1, c2, c3, x]
p1 = 1 + 3 x + 4 x^2; p2 = 3 + 10 x + 15 x^2; p3 = 3 + 11 x + 18 x^2;
Collect[c1 p1 + c2 p2 + c3 p3, x]
which gives
c1 + 3 c2 + 3 c3 + (3 c1 + 10 c2 + 11 c3) x + (4 c1 + 15 c2 + 18 c3) x^2
In the above, since the basis vectors of the polynominals are {1,x,x^2}, then see if there are c's, not all zero which makes the above sum zero. If you can find at least one c which makes the sum zero, then the polynomials are L.D. Else they are L.I.
So, write
eqs = {c1 + 3 c2 + 3 c3 == 0,
3 c1 + 10 c2 + 11 c3 == 0,
4 c1 + 15 c2 + 18 c3 == 0}
mat = {{1, 3, 3}, {3, 10, 11}, {4, 15, 18}}
Det[mat]
(* 0 *)
So there is
no solution. Since det is zero.
So the polynomials are L.I.
Another easier way to see this:
Solve[eqs, {c1, c2, c3}]
Will say there is no solution.
