Show that the Hermite polynomials form a basis of $\mathbb{P}_3$ I have this question that I took a shot at but I am not very familiar with Hermite or Laguerre, this my first time running across these type of polynomials and need some help please.
(a) The first four Hermite polynomials are $1, 2t,-2+4t^2,$ and $-12t+8t^3$. These polynomials arise naturally in the study of certain important differential equations in mathematical physics. Show that the first four Hermite polynomials form a basis of $\mathbb{P}_3$.
(b) The first four Laguerre polynomials are $1, 1-t, 2-4t + t^2,$ and $6-18t + 9t^2- t^3$. Show
that these four Laguerre polynomials form a basis of $\mathbb{P}_3$.
Results:
(a) The first four Hermite polynomials will be shown to form a basis of $\mathbb{P}_3$ by showing that they are linearly independent and that the number of polynomials equals the dimension of $\mathbb{P}_3$.
Consider the following linear combination of the four Hermite polynomials:
$x(1) + y(2t) + z(-2+4t^2) + w(-12t + 8t^3) = at^3 + bt^2 + ct + d$ 
The first four Hermite polynomials will be shown to be linearly independent by showing that the only linear combination of them that produces the zero polynomial ($0t^3+0t^2+0t+0$) is the trivial combination of zero times each polynomial.
That is all I could come about thus far with it. Can anyone refine or correct this if this is the wrong approach to the problem.

The only other thing to note, that Hermite $H_i$ and Laguerre $L_i$ polynomials are orthogonal.  In a sense that,
$\int H_i(x) H_j(x) w(x) dx = 0$ when $j \neq i$, where $w(x) = e^{x^2}$,
$\int L_i(x) L_j(x) v(x) dx = 0$ when $j \neq i$, where $v(x) = e^{-x}.$
