Rewrite $2b^{2}c^{2}+2c^{2}a^{2}+2a^{2}b^{2}-a^{4}-b^{4}-c^{4}$ as $\left(2ca\right)^{2}-\left(-b^{2}+c^{2}+a^{2}\right)^{2}$. The difference of squares can be factored using the rule: $p^{2}-q^{2}=\left(p-q\right)\left(p+q\right)$.
Consider $-a^{2}+2ac+b^{2}-c^{2}$. Rewrite $-a^{2}+2ac+b^{2}-c^{2}$ as $b^{2}-\left(c-a\right)^{2}$. The difference of squares can be factored using the rule: $p^{2}-q^{2}=\left(p-q\right)\left(p+q\right)$.
$$\left(b-c+a\right)\left(b+c-a\right)$$
Reorder the terms.
$$\left(a+b-c\right)\left(-a+b+c\right)$$
Consider $a^{2}+2ac-b^{2}+c^{2}$. Rewrite $a^{2}+2ac-b^{2}+c^{2}$ as $\left(c+a\right)^{2}-b^{2}$. The difference of squares can be factored using the rule: $p^{2}-q^{2}=\left(p-q\right)\left(p+q\right)$.
