Sum of Squares: Factoring Method

Problems that look like $(a \pm b)^2 + (c \pm d)^2$ can often be simplified by expanding the terms using FOIL. This avoids direct, cumbersome calculations. Usually, the numbers involved (like $a, b, c, d$) are chosen to be simple, such as multiples of 10 or 5.

For example, instead of calculating $19^2 + (10^2 - 9^2)$, you can recognize that $19 = 10+9$ and $10^2 - 9^2 = (10-9)(10+9)=19$. So the problem is $19^2+19$. An even better way is to expand $(10+9)^2 + (10^2-9^2)$, although this specific example in the text seems to have a typo. The general idea is to use algebraic identities to your advantage, such as $(a+b)^2 = a^2+2ab+b^2$ and difference of squares, to break down the problem into more manageable pieces.