Let's solve the expression $(1 - 4i)(-1 - 1i)$, where $i$ represents the imaginary unit.

We can apply the distributive property (FOIL method) to expand the expression:

$(1 - 4i)(-1 - 1i) = 1 \cdot (-1) + 1 \cdot (-1i) + (-4i) \cdot (-1) + (-4i) \cdot (-1i)$

Now, let's calculate each term:

1. $1 \cdot (-1) = -1$
2. $1 \cdot (-1i) = -i$
3. $(-4i) \cdot (-1) = 4i$
4. $(-4i) \cdot (-1i) = 4i^2 = 4(-1) = -4$ (since $i^2 = -1$)

Now, let's combine all the terms:

$-1 - i + 4i - 4$

Group the real and imaginary parts:

$(-1 - 4) + (-i + 4i) = -5 + 3i$

So, the result is:

$\boxed{-5 + 3i}$

Would you like more details on how this was done or have any other questions?

Here are 5 related questions to practice:

1. Simplify $(3 + 2i)(2 - i)$.
2. Expand $(5 - i)(-3 + 4i)$.
3. What is the conjugate of $2 + 3i$?
4. Find the magnitude of $4 + 7i$.
5. Simplify $(1 + i)^2$.

Tip: When multiplying complex numbers, remember to treat $i^2$ as $-1$.