Solve (2x-3)(x-4)-(4x^2-9)-(2x-5)^2

$$( 2 x - 3 ) ( x - 4 ) - ( 4 x ^ { 2 } - 9 ) - ( 2 x - 5 ) ^ { 2 }$$

Evaluate

$$-6x^{2}+9x-4$$

Solution Steps

Use the distributive property to multiply 2x-3 by x-4 and combine like terms.

$$2x^{2}-11x+12-\left(4x^{2}-9\right)-\left(2x-5\right)^{2}$$

To find the opposite of 4x^{2}-9, find the opposite of each term.

$$2x^{2}-11x+12-4x^{2}+9-\left(2x-5\right)^{2}$$

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

$$-2x^{2}-11x+12+9-\left(2x-5\right)^{2}$$

Add 12 and 9 to get 21.

$$-2x^{2}-11x+21-\left(2x-5\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-5\right)^{2}.

$$-2x^{2}-11x+21-\left(4x^{2}-20x+25\right)$$

To find the opposite of 4x^{2}-20x+25, find the opposite of each term.

$$-2x^{2}-11x+21-4x^{2}+20x-25$$

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

$$-6x^{2}-11x+21+20x-25$$

Combine -11x and 20x to get 9x.

$$-6x^{2}+9x+21-25$$

Subtract 25 from 21 to get -4.

$$-6x^{2}+9x-4$$

Expand

$$-6x^{2}+9x-4$$

Solution Steps

Use the distributive property to multiply 2x-3 by x-4 and combine like terms.

$$2x^{2}-11x+12-\left(4x^{2}-9\right)-\left(2x-5\right)^{2}$$

To find the opposite of 4x^{2}-9, find the opposite of each term.

$$2x^{2}-11x+12-4x^{2}+9-\left(2x-5\right)^{2}$$

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

$$-2x^{2}-11x+12+9-\left(2x-5\right)^{2}$$

Add 12 and 9 to get 21.

$$-2x^{2}-11x+21-\left(2x-5\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-5\right)^{2}.

$$-2x^{2}-11x+21-\left(4x^{2}-20x+25\right)$$

To find the opposite of 4x^{2}-20x+25, find the opposite of each term.

$$-2x^{2}-11x+21-4x^{2}+20x-25$$

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

$$-6x^{2}-11x+21+20x-25$$

Combine -11x and 20x to get 9x.

$$-6x^{2}+9x+21-25$$

Subtract 25 from 21 to get -4.

$$-6x^{2}+9x-4$$