Solve -m^4+5m^2n-6n^2-5m^2+12n-6

$$- m ^ { 4 } + 5 m ^ { 2 } n - 6 n ^ { 2 } - 5 m ^ { 2 } + 12 n - 6$$

Factor

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

Solution Steps

Consider -m^{4}+5m^{2}n-6n^{2}-5m^{2}+12n-6 as a polynomial over variable m.

$$-m^{4}+\left(5n-5\right)m^{2}-6n^{2}+12n-6$$

Find one factor of the form km^{p}+q, where km^{p} divides the monomial with the highest power -m^{4} and q divides the constant factor -6n^{2}+12n-6. One such factor is m^{2}-3n+3. Factor the polynomial by dividing it by this factor.

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

Evaluate

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