jmc

Since $f(x)$ has degree $2$, we know it is of the form $a{x}^2+bx+c$. A monic polynomial is one whose leading coefficient is $1$, so $a=1$. Since $f(0)=4$, we know $1(0{)}^2+b(0)+c=4$, so $c=4$. Since $f(1)=10$, we know $1(1{)}^2+b(1)+4=10$, so $b+5=10$ and $b=5$. Thus $f(x)=\boxed{x^2+5x+4}$.