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.