(1)

\begin{align*} a^{2}-b^{2} & =a^{2}-ab+ab-b^{2}\\ & =a\left(a-b\right)+b\left(a-b\right)\\ & =\left(a+b\right)\left(a-b\right) \end{align*}

(2)

\begin{align*} a^{2}\pm2ab+b^{2} & =a^{2}+b^{2}\pm2ab\\ & =\left(a\pm b\right)^{2}\mp2ab\pm2ab\\ & =\left(a\pm b\right)^{2} \end{align*}

(3)

\begin{align*} a^{2}+b^{2}+c^{2}+2\left(ab+bc+ca\right) & =a^{2}+2\left(b+c\right)a+b^{2}+c^{2}+2bc\\ & =a^{2}+2\left(b+c\right)a+\left(b+c\right)^{2}\\ & =\left(a+b+c\right)^{2} \end{align*}

(3)-2

(4)で\(n=3\)のときである。

(4)

\begin{align*} \sum_{j=1}^{n}\sum_{k=1}^{n}\left(a_{j}a_{k}\right) & =\left(\sum_{j=1}^{n}a_{j}\right)\left(\sum_{k=1}^{n}a_{k}\right)\\ & =\left(\sum_{k=1}^{n}a_{k}\right)^{2} \end{align*}