Prove a quadratic is always positive

$b^2-4ac=(-2)^2-4(1)(4)=-12<0$. Since the leading coefficient is $a=1>0$, the graph stays above the x-axis, so $y>0$ for every real $x$.

$y=(x-1)^2-1+4=(x-1)^2+3$. Since $(x-1)^2\ge0$, $y\ge3>0$.