Multiply (2 + 3i) by (4 - i). What is the product in standard form?

1. Multiply the two brackets the way two binomials are multiplied, first, outer, inner and last. 2. First gives 2 times 4, which is 8. 3. Outer gives 2 times negative i, which is negative 2i. 4. Inner gives 3i times 4, which is 12i. 5. Last gives 3i times negative i, which is negative 3 times i squared. 6. Since i squared is negative 1, that last piece becomes positive 3. 7. Collecting real parts gives 8 plus 3, which is 11, and collecting imaginary parts gives 10i. 8. The answer 8 - 3i comes from treating the brackets as if only first and last mattered. 9. The answer 5 + 10i comes from forgetting that i squared turns the last product positive. 10. The answer 11 - 10i comes from a sign slip when adding negative 2i to 12i. _Source: OpenStax Precalculus (CC BY 4.0), section 3.1 Complex Numbers_