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From Förberedande kurs i matematik 1

When the expression

(1+x+x2+x3)(2−x+x2+x4)

is expanded out, every term in the first bracket is multiplied by every term in the second bracket, i.e.

(1+x+x2+x3)(2−x+x2+x4)=12+1(−x)+1x2+1x4+x2+x(−x)+xx2+xx4+x22+x2(−x)+x2x2+x2x4+x32+x3(−x)+x3x2+x3x4.

If we only want to know the coefficient in front of x, we do not need to carry out the complete expansion of the expression; it is sufficient to find those combinations of a term from the first bracket and a term from the second bracket which, when multiplied, give an x-term. In this case, we have two such pairs: 1 multiplied by -x and x multiplied by 2,

(1+x+x2+x3)(2−x+x2+x4)=+1(−x)+x2+

so that the coefficient in front of x is −1+2=1.

We obtain the coefficient in front of x² by finding those combinations of a term from each bracket which give an x²-term; these are

(1+x+x2+x3)(2−x+x2+x4)=+1x2+x(−x)+x22+

The coefficient in front of x² is 1−1+2=2.