The grouping method
When the terms have no common factor, we use the grouping method. We split the terms into groups so that each group has a common factor and, after taking it out, the brackets are identical: ax + ay + bx + by = a(x + y) + b(x + y) = (x + y)(a + b). Now the common factor is the bracket (x + y) itself. We watch the signs when grouping: with a minus before a group, the signs of the terms put in brackets change, for example x² - xy - 3x + 3y = x(x - y) - 3(x - y). If the brackets do not come out identical, we try a different grouping or change a sign to make the brackets equal. The method rests on the associative and commutative laws of addition and the distributive law of multiplication.
“Parade of terms”: the four terms of a polynomial are on cards. Pupils split the cards into two groups, take the common factor out of each and show that identical brackets appear.