Solve x^{2}y+xy^2+2xyz+xy^2+y^3+2y^2z

$$x ^ { 2 } y + x y ^ { 2 } + 2 x y z + x y ^ { 2 } + y ^ { 3 } + 2 y ^ { 2 } z$$

Factor

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

Solution Steps

Factor out y.

$$y\left(x^{2}+xy+2xz+xy+y^{2}+2yz\right)$$

Consider x^{2}+xy+2xz+xy+y^{2}+2yz. Multiply and combine like terms.

$$x^{2}+2xy+2xz+y^{2}+2yz$$

Consider x^{2}+2xy+2xz+y^{2}+2yz. Consider x^{2}+2xy+2xz+y^{2}+2yz as a polynomial over variable x.

$$x^{2}+\left(2y+2z\right)x+y^{2}+2yz$$

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 y^{2}+2yz. One such factor is x+y. Factor the polynomial by dividing it by this factor.

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

Rewrite the complete factored expression.

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

Evaluate

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