Solve (4x+15)^2-(6x+1)^2+(x-7)*(x+1)

$$( 4 x + 15 ) ^ { 2 } - ( 6 x + 1 ) ^ { 2 } + ( x - 7 ) \times ( x + 1 )$$

Evaluate

$$\left(7-x\right)\left(19x+31\right)$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4x+15\right)^{2}.

$$16x^{2}+120x+225-\left(6x+1\right)^{2}+\left(x-7\right)\left(x+1\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(6x+1\right)^{2}.

$$16x^{2}+120x+225-\left(36x^{2}+12x+1\right)+\left(x-7\right)\left(x+1\right)$$

To find the opposite of 36x^{2}+12x+1, find the opposite of each term.

$$16x^{2}+120x+225-36x^{2}-12x-1+\left(x-7\right)\left(x+1\right)$$

Combine 16x^{2} and -36x^{2} to get -20x^{2}.

$$-20x^{2}+120x+225-12x-1+\left(x-7\right)\left(x+1\right)$$

Combine 120x and -12x to get 108x.

$$-20x^{2}+108x+225-1+\left(x-7\right)\left(x+1\right)$$

Subtract 1 from 225 to get 224.

$$-20x^{2}+108x+224+\left(x-7\right)\left(x+1\right)$$

Use the distributive property to multiply x-7 by x+1 and combine like terms.

$$-20x^{2}+108x+224+x^{2}-6x-7$$

Combine -20x^{2} and x^{2} to get -19x^{2}.

$$-19x^{2}+108x+224-6x-7$$

Combine 108x and -6x to get 102x.

$$-19x^{2}+102x+224-7$$

Subtract 7 from 224 to get 217.

$$-19x^{2}+102x+217$$

Expand

$$217+102x-19x^{2}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4x+15\right)^{2}.

$$16x^{2}+120x+225-\left(6x+1\right)^{2}+\left(x-7\right)\left(x+1\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(6x+1\right)^{2}.

$$16x^{2}+120x+225-\left(36x^{2}+12x+1\right)+\left(x-7\right)\left(x+1\right)$$

To find the opposite of 36x^{2}+12x+1, find the opposite of each term.

$$16x^{2}+120x+225-36x^{2}-12x-1+\left(x-7\right)\left(x+1\right)$$

Combine 16x^{2} and -36x^{2} to get -20x^{2}.

$$-20x^{2}+120x+225-12x-1+\left(x-7\right)\left(x+1\right)$$

Combine 120x and -12x to get 108x.

$$-20x^{2}+108x+225-1+\left(x-7\right)\left(x+1\right)$$

Subtract 1 from 225 to get 224.

$$-20x^{2}+108x+224+\left(x-7\right)\left(x+1\right)$$

Use the distributive property to multiply x-7 by x+1 and combine like terms.

$$-20x^{2}+108x+224+x^{2}-6x-7$$

Combine -20x^{2} and x^{2} to get -19x^{2}.

$$-19x^{2}+108x+224-6x-7$$

Combine 108x and -6x to get 102x.

$$-19x^{2}+102x+224-7$$

Subtract 7 from 224 to get 217.

$$-19x^{2}+102x+217$$