To determine the values for which the rational expression $\frac{x + 4}{x^2 + 5x + 6}$ is undefined, follow the procedure by setting the denominator equal to $0$ and solving for the variable $x$:

$x^2 + 5x + 6 = 0$

Factor the quadratic equation: $(x + 2)(x + 3) = 0$

Set each factor to zero: $x + 2 = 0$ or $x + 3 = 0$ $x = -2$ or $x = -3$

Thus, the rational expression is undefined for the values $x = -2$ and $x = -3$.