Solve -7m^{2}n+4n^{3}+m^{3}+6mn^{2}-n^3+m^3+7m^2n+5n^3

$$- 7 m ^ { 2 } n + 4 n ^ { 3 } + m ^ { 3 } + 6 m n ^ { 2 } - n ^ { 3 } + m ^ { 3 } + 7 m ^ { 2 } n + 5 n ^ { 3 }$$

Factor

$$2\left(m+n\right)\left(m^{2}-mn+4n^{2}\right)$$

Solution Steps

Multiply and combine like terms.

$$2m^{3}+6mn^{2}+8n^{3}$$

Factor out 2.

$$2\left(m^{3}+3mn^{2}+4n^{3}\right)$$

Consider m^{3}+3mn^{2}+4n^{3}. Consider m^{3}+3mn^{2}+4n^{3} as a polynomial over variable m.

$$m^{3}+3n^{2}m+4n^{3}$$

Find one factor of the form m^{k}+p, where m^{k} divides the monomial with the highest power m^{3} and p divides the constant factor 4n^{3}. One such factor is m+n. Factor the polynomial by dividing it by this factor.

$$\left(m+n\right)\left(m^{2}-mn+4n^{2}\right)$$

Rewrite the complete factored expression.

$$2\left(m+n\right)\left(m^{2}-mn+4n^{2}\right)$$

Evaluate

$$2\left(m^{3}+3mn^{2}+4n^{3}\right)$$