jmc

When graphed, this function is a parabola that opens upwards. Thus, the minimum possible value of y occurs at the vertex of the parabola. The $x$ coordinate of the vertex is $\frac{-b}{2a}$. Substituting the given values, this yields $\frac{-10}{2}=-5$. Substituting this for $x$ gives the minimum value of $y$ to be $$\begin{array}{rl}y& =x^2+10x+21\\ & =(-5{)}^2+10(-5)+21\\ & =25+(-50)+21\\ & =25-50+21\\ & =-25+21\\ & =\boxed{-4}\end{array}$$