State and prove the convolution theorem for Laplace transforms.

Use Laplace transforms to solve

$$2 f^{\prime}(t)-\int_{0}^{t}(t-\tau)^{2} f(\tau) d \tau=4 t H(t)$$

with $f(0)=0$, where $H(t)$ is the Heaviside function. You may assume that the Laplace transform, $\widehat{f}(s)$, of $f(t)$ exists for Re $s$ sufficiently large.