Solve (2x+1)(x-1)-(2x+1)^2-(2x+1)(2x-1)+6(x-1)

$$( 2 x + 1 ) ( x - 1 ) - ( 2 x + 1 ) ^ { 2 } - ( 2 x + 1 ) ( 2 x - 1 ) + 6 ( x - 1 )$$

Evaluate

$$-6x^{2}+x-7$$

Solution Steps

Use the distributive property to multiply 2x+1 by x-1 and combine like terms.

$$2x^{2}-x-1-\left(2x+1\right)^{2}-\left(2x+1\right)\left(2x-1\right)+6\left(x-1\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+1\right)^{2}.

$$2x^{2}-x-1-\left(4x^{2}+4x+1\right)-\left(2x+1\right)\left(2x-1\right)+6\left(x-1\right)$$

To find the opposite of 4x^{2}+4x+1, find the opposite of each term.

$$2x^{2}-x-1-4x^{2}-4x-1-\left(2x+1\right)\left(2x-1\right)+6\left(x-1\right)$$

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

$$-2x^{2}-x-1-4x-1-\left(2x+1\right)\left(2x-1\right)+6\left(x-1\right)$$

Combine -x and -4x to get -5x.

$$-2x^{2}-5x-1-1-\left(2x+1\right)\left(2x-1\right)+6\left(x-1\right)$$

Subtract 1 from -1 to get -2.

$$-2x^{2}-5x-2-\left(2x+1\right)\left(2x-1\right)+6\left(x-1\right)$$

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}. Square 1.

$$-2x^{2}-5x-2-\left(\left(2x\right)^{2}-1\right)+6\left(x-1\right)$$

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

$$-2x^{2}-5x-2-\left(2^{2}x^{2}-1\right)+6\left(x-1\right)$$

Calculate 2 to the power of 2 and get 4.

$$-2x^{2}-5x-2-\left(4x^{2}-1\right)+6\left(x-1\right)$$

To find the opposite of 4x^{2}-1, find the opposite of each term.

$$-2x^{2}-5x-2-4x^{2}+1+6\left(x-1\right)$$

Combine -2x^{2} and -4x^{2} to get -6x^{2}.

$$-6x^{2}-5x-2+1+6\left(x-1\right)$$

Add -2 and 1 to get -1.

$$-6x^{2}-5x-1+6\left(x-1\right)$$

Use the distributive property to multiply 6 by x-1.

$$-6x^{2}-5x-1+6x-6$$

Combine -5x and 6x to get x.

$$-6x^{2}+x-1-6$$

Subtract 6 from -1 to get -7.

$$-6x^{2}+x-7$$

Expand

$$-6x^{2}+x-7$$

Solution Steps

Use the distributive property to multiply 2x+1 by x-1 and combine like terms.

$$2x^{2}-x-1-\left(2x+1\right)^{2}-\left(2x+1\right)\left(2x-1\right)+6\left(x-1\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+1\right)^{2}.

$$2x^{2}-x-1-\left(4x^{2}+4x+1\right)-\left(2x+1\right)\left(2x-1\right)+6\left(x-1\right)$$

To find the opposite of 4x^{2}+4x+1, find the opposite of each term.

$$2x^{2}-x-1-4x^{2}-4x-1-\left(2x+1\right)\left(2x-1\right)+6\left(x-1\right)$$

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

$$-2x^{2}-x-1-4x-1-\left(2x+1\right)\left(2x-1\right)+6\left(x-1\right)$$

Combine -x and -4x to get -5x.

$$-2x^{2}-5x-1-1-\left(2x+1\right)\left(2x-1\right)+6\left(x-1\right)$$

Subtract 1 from -1 to get -2.

$$-2x^{2}-5x-2-\left(2x+1\right)\left(2x-1\right)+6\left(x-1\right)$$

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}. Square 1.

$$-2x^{2}-5x-2-\left(\left(2x\right)^{2}-1\right)+6\left(x-1\right)$$

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

$$-2x^{2}-5x-2-\left(2^{2}x^{2}-1\right)+6\left(x-1\right)$$

Calculate 2 to the power of 2 and get 4.

$$-2x^{2}-5x-2-\left(4x^{2}-1\right)+6\left(x-1\right)$$

To find the opposite of 4x^{2}-1, find the opposite of each term.

$$-2x^{2}-5x-2-4x^{2}+1+6\left(x-1\right)$$

Combine -2x^{2} and -4x^{2} to get -6x^{2}.

$$-6x^{2}-5x-2+1+6\left(x-1\right)$$

Add -2 and 1 to get -1.

$$-6x^{2}-5x-1+6\left(x-1\right)$$

Use the distributive property to multiply 6 by x-1.

$$-6x^{2}-5x-1+6x-6$$

Combine -5x and 6x to get x.

$$-6x^{2}+x-1-6$$

Subtract 6 from -1 to get -7.

$$-6x^{2}+x-7$$