Find all values of the variable \( x \) for which the rational expression is undefined. \[ \frac{8 x+19}{x^{2}+12 x+32} \] The expression is undefined when \( x= \)

To find the values of \( x \) for which the rational expression is undefined, we need to determine when the denominator is equal to zero. Set the denominator equal to zero: \[ x^{2} + 12x + 32 = 0 \] Next, we can factor this quadratic equation. We look for two numbers that multiply to \( 32 \) and add up to \( 12 \). These numbers are \( 4 \) and \( 8 \): \[ (x + 4)(x + 8) = 0 \] Setting each factor equal to zero gives us the values of \( x \): \[ x + 4 = 0 \quad \Rightarrow \quad x = -4 \] \[ x + 8 = 0 \quad \Rightarrow \quad x = -8 \] Therefore, the expression is undefined when \( x = -4 \) or \( x = -8 \). In conclusion, the rational expression is undefined at \( x = -4 \) and \( x = -8 \).