Solve 6x(7x-9)-(x+3)^2-(2x-4)(2x+4)

$$6 x ( 7 x - 9 ) - ( x + 3 ) ^ { 2 } - ( 2 x - 4 ) ( 2 x + 4 )$$

Evaluate

$$37x^{2}-60x+7$$

Solution Steps

Use the distributive property to multiply 6x by 7x-9.

$$42x^{2}-54x-\left(x+3\right)^{2}-\left(2x-4\right)\left(2x+4\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$42x^{2}-54x-\left(x^{2}+6x+9\right)-\left(2x-4\right)\left(2x+4\right)$$

To find the opposite of x^{2}+6x+9, find the opposite of each term.

$$42x^{2}-54x-x^{2}-6x-9-\left(2x-4\right)\left(2x+4\right)$$

Combine 42x^{2} and -x^{2} to get 41x^{2}.

$$41x^{2}-54x-6x-9-\left(2x-4\right)\left(2x+4\right)$$

Combine -54x and -6x to get -60x.

$$41x^{2}-60x-9-\left(2x-4\right)\left(2x+4\right)$$

Consider \left(2x-4\right)\left(2x+4\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 4.

$$41x^{2}-60x-9-\left(\left(2x\right)^{2}-16\right)$$

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

$$41x^{2}-60x-9-\left(2^{2}x^{2}-16\right)$$

Calculate 2 to the power of 2 and get 4.

$$41x^{2}-60x-9-\left(4x^{2}-16\right)$$

To find the opposite of 4x^{2}-16, find the opposite of each term.

$$41x^{2}-60x-9-4x^{2}+16$$

Combine 41x^{2} and -4x^{2} to get 37x^{2}.

$$37x^{2}-60x-9+16$$

Add -9 and 16 to get 7.

$$37x^{2}-60x+7$$

Expand

$$37x^{2}-60x+7$$

Solution Steps

Use the distributive property to multiply 6x by 7x-9.

$$42x^{2}-54x-\left(x+3\right)^{2}-\left(2x-4\right)\left(2x+4\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$42x^{2}-54x-\left(x^{2}+6x+9\right)-\left(2x-4\right)\left(2x+4\right)$$

To find the opposite of x^{2}+6x+9, find the opposite of each term.

$$42x^{2}-54x-x^{2}-6x-9-\left(2x-4\right)\left(2x+4\right)$$

Combine 42x^{2} and -x^{2} to get 41x^{2}.

$$41x^{2}-54x-6x-9-\left(2x-4\right)\left(2x+4\right)$$

Combine -54x and -6x to get -60x.

$$41x^{2}-60x-9-\left(2x-4\right)\left(2x+4\right)$$

Consider \left(2x-4\right)\left(2x+4\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 4.

$$41x^{2}-60x-9-\left(\left(2x\right)^{2}-16\right)$$

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

$$41x^{2}-60x-9-\left(2^{2}x^{2}-16\right)$$

Calculate 2 to the power of 2 and get 4.

$$41x^{2}-60x-9-\left(4x^{2}-16\right)$$

To find the opposite of 4x^{2}-16, find the opposite of each term.

$$41x^{2}-60x-9-4x^{2}+16$$

Combine 41x^{2} and -4x^{2} to get 37x^{2}.

$$37x^{2}-60x-9+16$$

Add -9 and 16 to get 7.

$$37x^{2}-60x+7$$