Solve 6(x+y)^2-5(x^2-y^2)-6(x-y)^2

$$6 ( x + y ) ^ { 2 } - 5 ( x ^ { 2 } - y ^ { 2 } ) - 6 ( x - y ) ^ { 2 }$$

Evaluate

$$\left(5y-x\right)\left(5x+y\right)$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+y\right)^{2}.

$$6\left(x^{2}+2xy+y^{2}\right)-5\left(x^{2}-y^{2}\right)-6\left(x-y\right)^{2}$$

Use the distributive property to multiply 6 by x^{2}+2xy+y^{2}.

$$6x^{2}+12xy+6y^{2}-5\left(x^{2}-y^{2}\right)-6\left(x-y\right)^{2}$$

Use the distributive property to multiply -5 by x^{2}-y^{2}.

$$6x^{2}+12xy+6y^{2}-5x^{2}+5y^{2}-6\left(x-y\right)^{2}$$

Combine 6x^{2} and -5x^{2} to get x^{2}.

$$x^{2}+12xy+6y^{2}+5y^{2}-6\left(x-y\right)^{2}$$

Combine 6y^{2} and 5y^{2} to get 11y^{2}.

$$x^{2}+12xy+11y^{2}-6\left(x-y\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-y\right)^{2}.

$$x^{2}+12xy+11y^{2}-6\left(x^{2}-2xy+y^{2}\right)$$

Use the distributive property to multiply -6 by x^{2}-2xy+y^{2}.

$$x^{2}+12xy+11y^{2}-6x^{2}+12xy-6y^{2}$$

Combine x^{2} and -6x^{2} to get -5x^{2}.

$$-5x^{2}+12xy+11y^{2}+12xy-6y^{2}$$

Combine 12xy and 12xy to get 24xy.

$$-5x^{2}+24xy+11y^{2}-6y^{2}$$

Combine 11y^{2} and -6y^{2} to get 5y^{2}.

$$-5x^{2}+24xy+5y^{2}$$

Expand

$$5y^{2}+24xy-5x^{2}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+y\right)^{2}.

$$6\left(x^{2}+2xy+y^{2}\right)-5\left(x^{2}-y^{2}\right)-6\left(x-y\right)^{2}$$

Use the distributive property to multiply 6 by x^{2}+2xy+y^{2}.

$$6x^{2}+12xy+6y^{2}-5\left(x^{2}-y^{2}\right)-6\left(x-y\right)^{2}$$

Use the distributive property to multiply -5 by x^{2}-y^{2}.

$$6x^{2}+12xy+6y^{2}-5x^{2}+5y^{2}-6\left(x-y\right)^{2}$$

Combine 6x^{2} and -5x^{2} to get x^{2}.

$$x^{2}+12xy+6y^{2}+5y^{2}-6\left(x-y\right)^{2}$$

Combine 6y^{2} and 5y^{2} to get 11y^{2}.

$$x^{2}+12xy+11y^{2}-6\left(x-y\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-y\right)^{2}.

$$x^{2}+12xy+11y^{2}-6\left(x^{2}-2xy+y^{2}\right)$$

Use the distributive property to multiply -6 by x^{2}-2xy+y^{2}.

$$x^{2}+12xy+11y^{2}-6x^{2}+12xy-6y^{2}$$

Combine x^{2} and -6x^{2} to get -5x^{2}.

$$-5x^{2}+12xy+11y^{2}+12xy-6y^{2}$$

Combine 12xy and 12xy to get 24xy.

$$-5x^{2}+24xy+11y^{2}-6y^{2}$$

Combine 11y^{2} and -6y^{2} to get 5y^{2}.

$$-5x^{2}+24xy+5y^{2}$$