Consider the following process:
$$ at+bW_t+atW_t+bW^{2}_t $$
For what a and b would this process be a Martingale?
I understand that this means we need to prove the following:
$$ E(at+bW_t+atW_t+bW^{2}_t|F_s) = as+bW_s+asW_s+bW^{2}_s $$
This is what I have done thus far:
$$ E[G_t|F_{s}]=E[at+bW_t+atW_t+bW^{2}_t|F_{s}] $$
$$ =at+bE[W_t|F_{s}]+atE[W_t|F_{s}]+bE[W^{2}_t|F_{s}] $$
$$ =at+bW_s+atW_s+bE[W^{2}_t|F_{s}] $$
Following guidance from Jose, I substituted $$ E(W^{2}_t-t|F_{s})=W^{2}_s-s $$
$$ E[G_t|F_{s}]=E[at+bW_t+atW_t+bW^{2}_t-bt+bt|F_{s}] $$ $$ = at+bW_s+atW_s+bt+bW^{2}_s-bs $$
We can therefore see that for $$ E[G_t|F_{s}]=Z_s, at+atW_s+bt=0. $$ $$ t(a+aW_s+b)=0 $$ $$ \therefore -a(1+W_s)=b $$
Is this correct? And is this a unique answer?
You are almost there; you need to find values of $a,b$ such that $-a(1+W_s) = b$ for all $s \geq 0$. The left-hand side is non-constant unless $a = 0$, which forces $b = 0$. This is the only solution.