Assuming that $\{ W ( t ) | t \geq 0 \}$ is a Brownian motion, I'm trying to check whether the process $$X ( t ) = W ( t ) + 4 t$$ is martingale with respect to filter $\mathcal{F}_t$. For this we should check if $\mathbb{E}[X(t)|\mathcal{F}_s] = X(s)$.
My try: \begin{align*} \mathbb{E}[W(t) + 4t|\mathcal{F}_s] &= \mathbb{E}[(W(t) - W(s) + W(s)) + 4t]\\ &= \mathbb{E}[(W(t) - W(s))] + \mathbb{E}[W(s)] + \mathbb{E}[4t]\\ &= t - s + W(s) + 4t\\ &= W(s) - s + 5t\\ \end{align*} Then since $W(s) - s + 5t \neq W(s) + 4s$, this is not martingale.
First, I was wondering if the approach is correct. Secondly, a good number of online examples are using Itô's formula to check for martingale. I was wondering if there is a measure by which we choose what way to solve it or it is just to see if we can find the expectation easily or not.