A rational expression is undefined when its denominator is equal to zero. In this case, the rational expression is given by:
\[ \frac{5x + 15}{-2x^2 - 16x - 32} \]
To find the values of \( x \) that make this expression undefined, we need to set the denominator equal to zero and solve for \( x \):
\[ -2x^2 - 16x - 32 = 0 \]
Let's simplify by dividing the entire equation by -2:
\[ x^2 + 8x + 16 = 0 \]
Next, we can factor the quadratic equation:
\[ (x + 4)(x + 4) = 0 \]
or, equivalently,
\[ (x + 4)^2 = 0 \]
Setting the factor equal to zero gives:
\[ x + 4 = 0 \implies x = -4 \]
Thus, the rational expression is undefined at:
\[ \boxed{-4} \]