Find the product of \( -4-i \) and its conjugate

To find the product of \( -4 - i \) and its conjugate, we first need to determine the conjugate. The conjugate of \( -4 - i \) is \( -4 + i \). Now we can multiply these two complex numbers: \[ (-4 - i)(-4 + i) \] Using the distributive property (or FOIL), we calculate: \[ (-4)(-4) + (-4)(i) + (-i)(-4) + (-i)(i) = 16 - 4i + 4i - i^2 \] The \( -4i \) and \( +4i \) cancel each other out: \[ 16 - i^2 \] Since \( i^2 = -1 \), we substitute: \[ 16 - (-1) = 16 + 1 = 17 \] Thus, the product of \( -4 - i \) and its conjugate is \( \boxed{17} \).