Solve 2ax^2+bx^2-2ay^2-by^2

$$2 a x ^ { 2 } + b x ^ { 2 } - 2 a y ^ { 2 } - b y ^ { 2 }$$

Factor

$$\left(x-y\right)\left(b+2a\right)\left(x+y\right)$$

Solution Steps

Do the grouping 2ax^{2}+bx^{2}-2ay^{2}-by^{2}=\left(2ax^{2}+bx^{2}\right)+\left(-2ay^{2}-by^{2}\right), and factor out x^{2} in the first and -y^{2} in the second group.

$$x^{2}\left(2a+b\right)-y^{2}\left(2a+b\right)$$

Factor out common term 2a+b by using distributive property.

$$\left(2a+b\right)\left(x^{2}-y^{2}\right)$$

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

$$\left(x-y\right)\left(x+y\right)$$

Rewrite the complete factored expression.

$$\left(2a+b\right)\left(x-y\right)\left(x+y\right)$$

Evaluate

$$\left(b+2a\right)\left(x^{2}-y^{2}\right)$$