Solve {l}{a^{2}+b^{2}-c^{2}-2ab}{3a^{2}+4ab+b^{2}-2ac-c^2}{3a^2-b^2-c^2+2ab-2bc+2c}

$$\left. \begin{array} { l } { a ^ { 2 } + b ^ { 2 } - c ^ { 2 } - 2 a b } \\ { 3 a ^ { 2 } + 4 a b + b ^ { 2 } - 2 a c - c ^ { 2 } } \\ { 3 a ^ { 2 } - b ^ { 2 } - c ^ { 2 } + 2 a b - 2 b c + 2 c } \end{array} \right.$$

Least Common Multiple

$$lcm(\left(b-a\right)^{2}-c^{2},\left(a+b-c\right)\left(3a+b+c\right),-\left(b+c\right)^{2}+2ab+3a^{2}+2c)$$

Solution Steps

Factor the expressions that are not already factored.

$$\left(-a+b\right)^{2}-c^{2}=\left(-a+b+c\right)\left(-a+b-c\right)$$

Identify all the factors and their highest power in all expressions. Multiply the highest powers of these factors to get the least common multiple.

$$\left(a-b-c\right)\left(a+b-c\right)\left(a-b+c\right)\left(3a+b+c\right)\left(-3a^{2}-2ab+b^{2}+2bc+c^{2}-2c\right)$$

Expand the expression.

$$-9a^{6}-8a^{3}c^{3}-11a^{2}b^{4}+10a^{2}b^{2}c^{2}-7a^{2}c^{4}+8a^{2}c^{3}-4ab^{2}c^{3}+4ab^{2}c^{2}+4abc^{3}+2ac^{5}-4ac^{4}+2acb^{4}-4acb^{3}+b^{6}-4b^{3}c^{3}+19b^{2}a^{4}-b^{2}c^{4}+4b^{2}c^{3}-8ba^{2}c^{3}-8ba^{2}c^{2}+2bc^{5}+8bc^{2}a^{3}-2bca^{4}+4bca^{3}+c^{6}-2c^{5}+15c^{2}a^{4}+4c^{2}a^{3}-c^{2}b^{4}+6ca^{5}-6ca^{4}+8ca^{2}b^{2}+2cb^{5}-2cb^{4}-8cb^{2}a^{3}$$

Evaluate

$$\left(b-a\right)^{2}-c^{2},\ \left(a+b-c\right)\left(3a+b+c\right),\ -\left(b+c\right)^{2}+2ab+3a^{2}+2c$$