Solve {l}{(3a^{2}-2b^{2}+c^{2})+(-2a^{2}+b^{2}-3c^{2})}{+(-2c^{2}+3b^{2}-a^2)}{-(2b^2+4c^2+a^2)}

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

Least Common Multiple

$$\left(-a^{2}+3b^{2}-2c^{2}\right)\left(a^{2}-b^{2}-2c^{2}\right)\left(a^{2}+2b^{2}+4c^{2}\right)$$

Solution Steps

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

$$\left(-a^{2}+3b^{2}-2c^{2}\right)\left(a^{2}-b^{2}-2c^{2}\right)\left(a^{2}+2b^{2}+4c^{2}\right)$$

Expand the expression.

$$-a^{6}+5a^{2}b^{4}+12a^{2}b^{2}c^{2}+4a^{2}c^{4}-6b^{6}+2b^{2}a^{4}-8b^{2}c^{4}+16c^{6}-4c^{2}a^{4}-20c^{2}b^{4}$$

Evaluate

$$a^{2}-b^{2}-2c^{2},\ -a^{2}+3b^{2}-2c^{2},\ -\left(a^{2}+2b^{2}+4c^{2}\right)$$