math help i dont know what I am doing

Question

Evaluate the expression (1−4i)(−2−1i)(1-4i)(-2-1i) and write the result in the form a+bia+bi.

The real number aa equals   

The real number bb equals   

Answer 1

The key to multiplying two complex numbers in the form (a+bi) is realizing that i²=-1.

Doing the multiplication is just like using FOIL when multiplying two binomials.

So:

The product of the 1st terms is : (1)(-2) = -2

The outer product is: (-1i)(1) = -1i

The inner product is: (-4i)(-2) = 8i

And finally the product of the last terms is: (-4i)(-1i) = +4i², but replacing the i² with -1 results in -4.

Here is what it might look like if you did this on scrap paper:

-2 - 1i +8i -4 = -6+7i

With some practice you can probably reach the result mentally, i.e. without writing down intermediate steps.

In this case a= -6 and b = 7 (assuming no careless mistakes were made)