Find the monic quadratic polynomial, in $x,$ with real coefficients, which has $-2 - i \sqrt{5}$ as a root.
If a polynomial has real coefficients, then any complex conjugate of a root must also be a root.  Hence, the other root is $-2 + i \sqrt{5}.$  Thus, the polynomial is
\[(x + 2 + i \sqrt{5})(x + 2 - i \sqrt{5}) = (x + 2)^2 - 5i^2 = \boxed{x^2 + 4x + 9}.\]