Generally, Hermite polynomials can be described using the Rodrigues formula:
$$ H_n(x) = (-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2} $$
And the first few polynomials (for $n = 0,1,2,3,4,5...$) are well known:
$$ H_0(x) = 1\\ H_1(x) = 2x\\ H_2(x) = 4x^2 - 2\\ H_3(x) = 8x^3 - 12x\\ H_4(x) = 16x^4 - 48x^2 + 12\\ H_5(x) = 32x^5 - 160x^3 + 120x\\ $$
I am asked to derive two "ladder operators", $L_{\pm}$, (like the ones used for the harmonic oscillator in quantum mechanics), such that
$$ L_{\pm}H_{n}(x) = H_{n\pm 1}(x) $$
It seems the $L_{+}$ would be something like "multiply $2x$ each time and do other stuff" -- or more eloquently, "multiply $\frac{d}{dx} e^{x^2}$ and ..." -- but I'm having trouble getting a full answer.