Use Duhamel’s principle and the heat kernel to solve the initial value problem

Question

Use Duhamel’s principle and the heat kernel to solve the initial value problem

$u_t(x, t) − ku_{xx}(x, t) = xte^{−t^2}$
$∀x ∈ \mathbb{R}, t > 0$

subject to-

$u(x, 0) = x^2 − 3x − 1$

$ ∀x ∈ \mathbb{R}.$

Im very unsure whenever I get a Duhamel’s principle question so any help will be appreciated