Solve x^2-8y^2+2z^2+2xy+3xz

$${ x }^{ 2 } -8 { y }^{ 2 } +2 { z }^{ 2 } +2xy+3xz$$

Factor

$$\left(x-2y+z\right)\left(x+4y+2z\right)$$

Solution Steps

Consider x^{2}-8y^{2}+2z^{2}+2xy+3xz as a polynomial over variable x.

$$x^{2}+\left(2y+3z\right)x-8y^{2}+2z^{2}$$

Find one factor of the form x^{k}+m, where x^{k} divides the monomial with the highest power x^{2} and m divides the constant factor -8y^{2}+2z^{2}. One such factor is x+4y+2z. Factor the polynomial by dividing it by this factor.

$$\left(x+4y+2z\right)\left(x-2y+z\right)$$

Evaluate

$$\left(x-2y+z\right)\left(x+4y+2z\right)$$