Solve left(-2x+1right)^2+xleft(x-2right)+left(x+2right)left(x-3right)-left(3-xright)left(3+xright)

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

Evaluate

$$7\left(x-2\right)\left(x+1\right)$$

Solution Steps

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

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

Use the distributive property to multiply x by x-2.

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

Combine 4x^{2} and x^{2} to get 5x^{2}.

$$5x^{2}-4x+1-2x+\left(x+2\right)\left(x-3\right)-\left(3-x\right)\left(3+x\right)$$

Combine -4x and -2x to get -6x.

$$5x^{2}-6x+1+\left(x+2\right)\left(x-3\right)-\left(3-x\right)\left(3+x\right)$$

Use the distributive property to multiply x+2 by x-3 and combine like terms.

$$5x^{2}-6x+1+x^{2}-x-6-\left(3-x\right)\left(3+x\right)$$

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

$$6x^{2}-6x+1-x-6-\left(3-x\right)\left(3+x\right)$$

Combine -6x and -x to get -7x.

$$6x^{2}-7x+1-6-\left(3-x\right)\left(3+x\right)$$

Subtract 6 from 1 to get -5.

$$6x^{2}-7x-5-\left(3-x\right)\left(3+x\right)$$

Consider \left(3-x\right)\left(3+x\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 3.

$$6x^{2}-7x-5-\left(9-x^{2}\right)$$

To find the opposite of 9-x^{2}, find the opposite of each term.

$$6x^{2}-7x-5-9+x^{2}$$

Subtract 9 from -5 to get -14.

$$6x^{2}-7x-14+x^{2}$$

Combine 6x^{2} and x^{2} to get 7x^{2}.

$$7x^{2}-7x-14$$

Expand

$$7x^{2}-7x-14$$

Solution Steps

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

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

Use the distributive property to multiply x by x-2.

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

Combine 4x^{2} and x^{2} to get 5x^{2}.

$$5x^{2}-4x+1-2x+\left(x+2\right)\left(x-3\right)-\left(3-x\right)\left(3+x\right)$$

Combine -4x and -2x to get -6x.

$$5x^{2}-6x+1+\left(x+2\right)\left(x-3\right)-\left(3-x\right)\left(3+x\right)$$

Use the distributive property to multiply x+2 by x-3 and combine like terms.

$$5x^{2}-6x+1+x^{2}-x-6-\left(3-x\right)\left(3+x\right)$$

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

$$6x^{2}-6x+1-x-6-\left(3-x\right)\left(3+x\right)$$

Combine -6x and -x to get -7x.

$$6x^{2}-7x+1-6-\left(3-x\right)\left(3+x\right)$$

Subtract 6 from 1 to get -5.

$$6x^{2}-7x-5-\left(3-x\right)\left(3+x\right)$$

Consider \left(3-x\right)\left(3+x\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 3.

$$6x^{2}-7x-5-\left(9-x^{2}\right)$$

To find the opposite of 9-x^{2}, find the opposite of each term.

$$6x^{2}-7x-5-9+x^{2}$$

Subtract 9 from -5 to get -14.

$$6x^{2}-7x-14+x^{2}$$

Combine 6x^{2} and x^{2} to get 7x^{2}.

$$7x^{2}-7x-14$$