To multiply two co mplex numbers, use the Distributive Property, or the FOIL method, just as you do when multiplying real numbers or algebraic expressions. A) 4i(5-3i) B) (1+4i)(2-7i) C) (2-9i)^2

## Step 1: Distribute 4i in A<br />### We distribute $4i$ to both terms inside the parenthesis: $4i(5-3i) = 4i(5) - 4i(3i) = 20i - 12i^2$. Since $i^2 = -1$, we have $20i - 12(-1) = 12 + 20i$.<br /><br />## Step 2: FOIL method for B<br />### We use the FOIL method to expand the expression: $(1+4i)(2-7i) = 1(2) + 1(-7i) + 4i(2) + 4i(-7i) = 2 - 7i + 8i - 28i^2 = 2 + i - 28(-1) = 2 + i + 28 = 30 + i$.<br /><br />## Step 3: Expand the square in C<br />### We expand the square: $(2-9i)^2 = (2-9i)(2-9i) = 2(2) + 2(-9i) - 9i(2) - 9i(-9i) = 4 - 18i - 18i + 81i^2 = 4 - 36i + 81(-1) = 4 - 36i - 81 = -77 - 36i$.