FOIL is a mnemonic rule for multiplying binomial (that is, two-term) algebraic expressions. FOIL abbreviates the sequence "First, Outside, Inside, Last"; it's a way of remembering that the product is the sum of the products of those four combinations of terms.
For instance, if we multiply the two expressions (x + 1) (x + 2) then the result is the sum of these four products: x times x (the First terms of each expression) x times 2 (the Outside pair of terms) 1 times x (the Inside pair of terms) 1 times 2 (the Last terms of each expression) and so (x + 1) (x + 2) = x^2 + 2x + 1x + 2 = x^2 + 3x + 2 [where the ^ is the usual way we indicate exponents here in Answers, because they're hard to represent in an online text environment].
Now, compare this to multiplying a pair of two-digit integers: 37 × 43 = (30 × 40) + (30 × 3) + (7 × 40) + (7 × 3) = 1200 + 90 + 280 + 21 = 1591
The reason the two processes resemble each other is that multiplication is multiplication; the difference in the ways we represent the factors doesn't make it a fundamentally different operation.
