Solve (sqrt{2x}+sqrt{3y})(sqrt{2x}-sqrt{3y})-(x+y+1)^2

$$( \sqrt { 2 x } + \sqrt { 3 y } ) ( \sqrt { 2 x } - \sqrt { 3 y } ) - ( x + y + 1 ) ^ { 2 }$$

Evaluate

$$-\left(x+y+1\right)^{2}+2x-3y$$

Solution Steps

Consider \left(\sqrt{2x}+\sqrt{3y}\right)\left(\sqrt{2x}-\sqrt{3y}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(\sqrt{2x}\right)^{2}-\left(\sqrt{3y}\right)^{2}-\left(x+y+1\right)^{2}$$

Calculate \sqrt{2x} to the power of 2 and get 2x.

$$2x-\left(\sqrt{3y}\right)^{2}-\left(x+y+1\right)^{2}$$

Calculate \sqrt{3y} to the power of 2 and get 3y.

$$2x-3y-\left(x+y+1\right)^{2}$$

Square x+y+1.

$$2x-3y-\left(x^{2}+2xy+2x+y^{2}+2y+1\right)$$

To find the opposite of x^{2}+2xy+2x+y^{2}+2y+1, find the opposite of each term.

$$2x-3y-x^{2}-2xy-2x-y^{2}-2y-1$$

Combine 2x and -2x to get 0.

$$-3y-x^{2}-2xy-y^{2}-2y-1$$

Combine -3y and -2y to get -5y.

$$-5y-x^{2}-2xy-y^{2}-1$$

Differentiate w.r.t. x

$$-2x-2y$$