Solve (x+2y)(x-2y)+(x+1)(x-1)

$$( x + 2 y ) ( x - 2 y ) + ( x + 1 ) ( x - 1 )$$

Evaluate

$$2x^{2}-4y^{2}-1$$

Solution Steps

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

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

Expand \left(2y\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

$$x^{2}-4y^{2}+\left(x+1\right)\left(x-1\right)$$

Consider \left(x+1\right)\left(x-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$x^{2}-4y^{2}+x^{2}-1^{2}$$

Calculate 1 to the power of 2 and get 1.

$$x^{2}-4y^{2}+x^{2}-1$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}-4y^{2}-1$$

Expand

$$2x^{2}-4y^{2}-1$$

Solution Steps

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

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

Expand \left(2y\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

$$x^{2}-4y^{2}+\left(x+1\right)\left(x-1\right)$$

Consider \left(x+1\right)\left(x-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$x^{2}-4y^{2}+x^{2}-1^{2}$$

Calculate 1 to the power of 2 and get 1.

$$x^{2}-4y^{2}+x^{2}-1$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}-4y^{2}-1$$