The number of quadratic expressions with the coefficients drawn from the set $\left\{0,1,2,3\right\}$  is

$a{x}^2+bx+cwherea\ne 0$

$\underset{\_}{3}\underset{\_}{4}\underset{\_}{4}=3\times 4\times 4=48$

1^st place can be filled with $\left(1,2,3\right)$  in 3 ways

2^nd & 3^rd place can be filled in 4 ways $\left(0,1,2,3\right)$