To divide the complex numbers $\frac{1 - 6i}{-5 - 2i}$, we can follow the steps below to rationalize the denominator and simplify the expression:

Step 1: Multiply both numerator and denominator by the complex conjugate of the denominator.

The complex conjugate of $-5 - 2i$ is $-5 + 2i$. Multiplying by the conjugate will eliminate the imaginary part in the denominator.

$\frac{1 - 6i}{-5 - 2i} \times \frac{-5 + 2i}{-5 + 2i}$

Step 2: Perform the multiplication for both numerator and denominator.

Numerator:

$(1 - 6i)(-5 + 2i) = 1(-5) + 1(2i) + (-6i)(-5) + (-6i)(2i)$ Breaking this down: $1(-5) = -5, \quad 1(2i) = 2i, \quad (-6i)(-5) = 30i, \quad (-6i)(2i) = -12i^2$ Since $i^2 = -1$, we have: $-12i^2 = 12$ So, the numerator becomes: $-5 + 2i + 30i + 12 = 7 + 32i$

Denominator:

$(-5 - 2i)(-5 + 2i) = (-5)^2 - (2i)^2 = 25 - 4i^2$ Since $i^2 = -1$, this simplifies to: $25 - 4(-1) = 25 + 4 = 29$

Step 3: Write the result.

Now we can write the result as: $\frac{7 + 32i}{29}$ This is the simplified form of the division.

Step 4: Separate into real and imaginary parts (if needed):

$\frac{7}{29} + \frac{32}{29}i$

Thus, the final result is: $\frac{7}{29} + \frac{32}{29}i$

Let me know if you'd like more details or have questions!

Here are 5 related questions for further practice:

1. Divide $\frac{3 - 2i}{4 + 3i}$ and simplify.
2. Multiply $(1 + i)(1 - i)$.
3. Rationalize the denominator of $\frac{2}{1 + 3i}$.
4. Find the modulus of the complex number $7 + 32i$.
5. Express the complex number $\frac{5}{i}$ in standard form.

Tip: When multiplying complex numbers, remember that $i^2 = -1$, which helps in simplifying terms containing $i^2$.