From Förberedande kurs i matematik 1

When the expression \displaystyle \left( 1+x+x^{2}+x^{3} \right)\left( 2-x+x^{2}+x^{4} \right) is expanded out,

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

\displaystyle \begin{align} & \left( 1+x+x^{2}+x^{3} \right)\left( 2-x+x^{2}+x^{4} \right) \\ & =1\centerdot 2+1\centerdot \left( -x \right)+1\centerdot x^{2}+1\centerdot x^{4}+x\centerdot 2+x\centerdot \left( -x \right) \\ & +x\centerdot x^{2}+x\centerdot x^{4}+x^{2}\centerdot 2+x^{2}\centerdot \left( -x \right)+x^{2}\centerdot x^{2}+x^{2}\centerdot x^{4} \\ & +x^{3}\centerdot 2+x^{3}\centerdot \left( -x \right)+x^{3}\centerdot x^{2}+x^{3}\centerdot x^{4} \\ \end{align}

If we only want to know the coefficient in front of \displaystyle 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 \displaystyle x^{1} -term. In this case, we have two such pairs: \displaystyle 1 multiplied by - \displaystyle x and \displaystyle x multiplied by \displaystyle 2 ,

\displaystyle \begin{align} & \left( 1+x+x^{2}+x^{3} \right)\left( 2-x+x^{2}+x^{4} \right)=...+1\centerdot \left( -x \right)+x\centerdot 2+... \\ & \\ \end{align}

so that the coefficient in front of \displaystyle x is \displaystyle -1+2=1

We obtain the coefficient in front of \displaystyle x^{2} by finding those combinations of a term from each bracket which give an \displaystyle x^{2} -term; these are

\displaystyle \left( 1+x+x^{2}+x^{3} \right)\left( 2-x+x^{2}+x^{4} \right)=...+1\centerdot x^{2}+x\centerdot \left( -x \right)+x^{2}\centerdot 2+...

The coefficient in front of \displaystyle x^{2} is \displaystyle 1-1+2 .