Laplace transform of two arbitrary functions

Question

Sanity check needed - if I have the expression

$f(t) = x$

Then taking the Laplace transform of both sides yields

$\mathcal{L}\Big(f(t)\Big) = \mathcal{L}(x)$

Is it true that if I had another expression

$f(t)g(t) = x$

The Laplace transform of both sides would give?

$\mathcal{L}\Big(f(t)g(t)\Big) = \mathcal{L}(x)$

Answer 1

It seems correct. But note that we don't necessarily have $$\mathcal{L}(f(t)g(t))=\mathcal{L}(f(t))\times\mathcal{L}(g(t)) $$