Solve (2x+1)(2x-1)-(x+1)(3x-2)

$$( 2 x + 1 ) ( 2 x - 1 ) - ( x + 1 ) ( 3 x - 2 )$$

Evaluate

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

Solution Steps

Consider \left(2x+1\right)\left(2x-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}.

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

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

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

Calculate 2 to the power of 2 and get 4.

$$4x^{2}-1^{2}-\left(x+1\right)\left(3x-2\right)$$

Calculate 1 to the power of 2 and get 1.

$$4x^{2}-1-\left(x+1\right)\left(3x-2\right)$$

Apply the distributive property by multiplying each term of x+1 by each term of 3x-2.

$$4x^{2}-1-\left(3x^{2}-2x+3x-2\right)$$

Combine -2x and 3x to get x.

$$4x^{2}-1-\left(3x^{2}+x-2\right)$$

To find the opposite of 3x^{2}+x-2, find the opposite of each term.

$$4x^{2}-1-3x^{2}-x-\left(-2\right)$$

The opposite of -2 is 2.

$$4x^{2}-1-3x^{2}-x+2$$

Combine 4x^{2} and -3x^{2} to get x^{2}.

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

Add -1 and 2 to get 1.

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

Expand

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

Solution Steps

Consider \left(2x+1\right)\left(2x-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}.

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

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

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

Calculate 2 to the power of 2 and get 4.

$$4x^{2}-1^{2}-\left(x+1\right)\left(3x-2\right)$$

Calculate 1 to the power of 2 and get 1.

$$4x^{2}-1-\left(x+1\right)\left(3x-2\right)$$

Apply the distributive property by multiplying each term of x+1 by each term of 3x-2.

$$4x^{2}-1-\left(3x^{2}-2x+3x-2\right)$$

Combine -2x and 3x to get x.

$$4x^{2}-1-\left(3x^{2}+x-2\right)$$

To find the opposite of 3x^{2}+x-2, find the opposite of each term.

$$4x^{2}-1-3x^{2}-x-\left(-2\right)$$

The opposite of -2 is 2.

$$4x^{2}-1-3x^{2}-x+2$$

Combine 4x^{2} and -3x^{2} to get x^{2}.

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

Add -1 and 2 to get 1.

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