Regroup terms.
\[{(a-b)}^{2}-2{c}^{prime}{a}^{2}-{(b+c)}^{2}+{(b-c)}^{2}-{a}^{2}{b}^{2}-{(c+a)}^{2}+{(c-a)}^{2}-b{c}^{2}-{(a+b)}^{2}\]
Use Square of Difference: \({(a-b)}^{2}={a}^{2}-2ab+{b}^{2}\).
\[{a}^{2}-2ab+{b}^{2}-2{c}^{prime}{a}^{2}-{(b+c)}^{2}+{(b-c)}^{2}-{a}^{2}{b}^{2}-{(c+a)}^{2}+{(c-a)}^{2}-b{c}^{2}-{(a+b)}^{2}\]
Use Square of Sum: \({(a+b)}^{2}={a}^{2}+2ab+{b}^{2}\).
\[{a}^{2}-2ab+{b}^{2}-2{c}^{prime}{a}^{2}-({b}^{2}+2bc+{c}^{2})+{(b-c)}^{2}-{a}^{2}{b}^{2}-{(c+a)}^{2}+{(c-a)}^{2}-b{c}^{2}-{(a+b)}^{2}\]
Use Square of Difference: \({(a-b)}^{2}={a}^{2}-2ab+{b}^{2}\).
\[{a}^{2}-2ab+{b}^{2}-2{c}^{prime}{a}^{2}-({b}^{2}+2bc+{c}^{2})+{b}^{2}-2bc+{c}^{2}-{a}^{2}{b}^{2}-{(c+a)}^{2}+{(c-a)}^{2}-b{c}^{2}-{(a+b)}^{2}\]
Use Square of Sum: \({(a+b)}^{2}={a}^{2}+2ab+{b}^{2}\).
\[{a}^{2}-2ab+{b}^{2}-2{c}^{prime}{a}^{2}-({b}^{2}+2bc+{c}^{2})+{b}^{2}-2bc+{c}^{2}-{a}^{2}{b}^{2}-({c}^{2}+2ca+{a}^{2})+{(c-a)}^{2}-b{c}^{2}-{(a+b)}^{2}\]
Use Square of Difference: \({(a-b)}^{2}={a}^{2}-2ab+{b}^{2}\).
\[{a}^{2}-2ab+{b}^{2}-2{c}^{prime}{a}^{2}-({b}^{2}+2bc+{c}^{2})+{b}^{2}-2bc+{c}^{2}-{a}^{2}{b}^{2}-({c}^{2}+2ca+{a}^{2})+{c}^{2}-2ca+{a}^{2}-b{c}^{2}-{(a+b)}^{2}\]
Use Square of Sum: \({(a+b)}^{2}={a}^{2}+2ab+{b}^{2}\).
\[{a}^{2}-2ab+{b}^{2}-2{c}^{prime}{a}^{2}-({b}^{2}+2bc+{c}^{2})+{b}^{2}-2bc+{c}^{2}-{a}^{2}{b}^{2}-({c}^{2}+2ca+{a}^{2})+{c}^{2}-2ca+{a}^{2}-b{c}^{2}-({a}^{2}+2ab+{b}^{2})\]
Remove parentheses.
\[{a}^{2}-2ab+{b}^{2}-2{c}^{prime}{a}^{2}-{b}^{2}-2bc-{c}^{2}+{b}^{2}-2bc+{c}^{2}-{a}^{2}{b}^{2}-{c}^{2}-2ca-{a}^{2}+{c}^{2}-2ca+{a}^{2}-b{c}^{2}-{a}^{2}-2ab-{b}^{2}\]
Collect like terms.
\[({a}^{2}-{a}^{2}+{a}^{2}-{a}^{2})+(-2ab-2ab)+({b}^{2}-{b}^{2}+{b}^{2}-{b}^{2})-2{c}^{prime}{a}^{2}+(-2bc-2bc)+(-{c}^{2}+{c}^{2}-{c}^{2}+{c}^{2})-{a}^{2}{b}^{2}+(-2ca-2ca)-b{c}^{2}\]
Simplify.
\[-4ab-2{c}^{prime}{a}^{2}-4bc-{a}^{2}{b}^{2}-4ca-b{c}^{2}\]
-4*a*b-2*c^prime*a^2-4*b*c-a^2*b^2-4*c*a-b*c^2
