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a^{2}+\left(-2c-2b\right)a+b^{2}+2bc
Consider a^{2}+b^{2}+2bc-2ca-2ab as a polynomial over variable a.
\left(a-b\right)\left(a-b-2c\right)
Find one factor of the form a^{k}+m, where a^{k} divides the monomial with the highest power a^{2} and m divides the constant factor b^{2}+2bc. One such factor is a-b. Factor the polynomial by dividing it by this factor.