Solve 45am^2-20ax^2+20axy-5ay^2

$$45 a m ^ { 2 } - 20 a x ^ { 2 } + 20 a x y - 5 a y ^ { 2 }$$

Factor

$$5a\left(3m+y-2x\right)\left(2x-y+3m\right)$$

Solution Steps

Factor out 5.

$$5\left(9am^{2}-4ax^{2}+4axy-ay^{2}\right)$$

Consider 9am^{2}-4ax^{2}+4axy-ay^{2}. Factor out a.

$$a\left(9m^{2}-4x^{2}+4xy-y^{2}\right)$$

Consider 9m^{2}-4x^{2}+4xy-y^{2}. Rewrite as difference of two terms.

$$9m^{2}-\left(4x^{2}-4xy+y^{2}\right)$$

Rewrite the terms as squares.

$$\left(3m\right)^{2}-\left(2x-y\right)^{2}$$

The difference of squares can be factored using the rule: p^{2}-q^{2}=\left(p-q\right)\left(p+q\right).

$$\left(3m-2x+y\right)\left(3m+2x-y\right)$$

Reorder the terms.

$$\left(-2x+y+3m\right)\left(2x-y+3m\right)$$

Rewrite the complete factored expression.

$$5a\left(-2x+y+3m\right)\left(2x-y+3m\right)$$

Evaluate

$$-5a\left(y-2x\right)^{2}+45am^{2}$$