Multiplying two complex numbers is expanding two brackets, with one extra step at the end. These free printable worksheets give ten products like (2 − 2i)(4 + 2i), every part a whole number from −6 to 6, and the answer key writes out each line of the expansion.
If they get stuck
The answer key prints on a separate page.
Up to the last step this is (x + 3)(x − 2): four products, first, outer, inner and last. (2 − 2i)(4 + 2i) gives 8 + 4i − 8i − 4i². The difference is in the final term. With x, the x² stays where it is; with i, i² is −1, so −4i² is +4 — a real number that joins the 8 and leaves the imaginary terms alone. The answer is 12 − 4i.
That sign is the mistake the sheet is built around. The last product always carries an i², so its sign always turns over on the way into the real part: +18i² becomes −18, and −4i² becomes +4. Every key gives that step a line of its own, with the real numbers gathered at the front before the terms in i, so a wrong answer can be traced to the one term that kept its old sign.
(−3 + 3i)(−5 + 6i)