If $a\&b=a^2+b+1$ and a\ b = a + b^2 + 2$,thenfindthevalueof$ (4\$3)\&2$.

Let's solve the composite operations systematically step-by-step using BODMAS rules:
1. First, solve the inner bracket operation (4\ 3)$:$a\ b = a + b^2 + 2
$$

$$4\$3=4+(3{)}^2+2=4+9+2=15$$

2. Now, evaluate the outer operation $(15)\&2$:

$$a\&b=a^2+b+1$$

$$15\&2=(15{)}^2+2+1=225+2+1=228$$