Solve a^2-8b^2+2c^2+2ab+3ac=

$$a ^ { 2 } - 8 b ^ { 2 } + 2 c ^ { 2 } + 2 a b + 3 a c =$$

Factor

$$\left(a-2b+c\right)\left(a+4b+2c\right)$$

Solution Steps

Consider a^{2}-8b^{2}+2c^{2}+2ab+3ac as a polynomial over variable a.

$$a^{2}+\left(2b+3c\right)a-8b^{2}+2c^{2}$$

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 -8b^{2}+2c^{2}. One such factor is a+4b+2c. Factor the polynomial by dividing it by this factor.

$$\left(a+4b+2c\right)\left(a-2b+c\right)$$

Evaluate

$$\left(a-2b+c\right)\left(a+4b+2c\right)$$