Solve (3x^2+15)(2x-x^2)-(x^3+15x+1)(2-2x)

$$( 3 x ^ { 2 } + 15 ) ( 2 x - x ^ { 2 } ) - ( x ^ { 3 } + 15 x + 1 ) ( 2 - 2 x )$$

Evaluate

$$-x^{4}+4x^{3}+15x^{2}+2x-2$$

Solution Steps

Use the distributive property to multiply 3x^{2}+15 by 2x-x^{2}.

$$6x^{3}-3x^{4}+30x-15x^{2}-\left(x^{3}+15x+1\right)\left(2-2x\right)$$

Use the distributive property to multiply x^{3}+15x+1 by 2-2x and combine like terms.

$$6x^{3}-3x^{4}+30x-15x^{2}-\left(2x^{3}-2x^{4}+28x-30x^{2}+2\right)$$

To find the opposite of 2x^{3}-2x^{4}+28x-30x^{2}+2, find the opposite of each term.

$$6x^{3}-3x^{4}+30x-15x^{2}-2x^{3}+2x^{4}-28x+30x^{2}-2$$

Combine 6x^{3} and -2x^{3} to get 4x^{3}.

$$4x^{3}-3x^{4}+30x-15x^{2}+2x^{4}-28x+30x^{2}-2$$

Combine -3x^{4} and 2x^{4} to get -x^{4}.

$$4x^{3}-x^{4}+30x-15x^{2}-28x+30x^{2}-2$$

Combine 30x and -28x to get 2x.

$$4x^{3}-x^{4}+2x-15x^{2}+30x^{2}-2$$

Combine -15x^{2} and 30x^{2} to get 15x^{2}.

$$4x^{3}-x^{4}+2x+15x^{2}-2$$

Expand

$$-x^{4}+4x^{3}+15x^{2}+2x-2$$

Solution Steps

Use the distributive property to multiply 3x^{2}+15 by 2x-x^{2}.

$$6x^{3}-3x^{4}+30x-15x^{2}-\left(x^{3}+15x+1\right)\left(2-2x\right)$$

Use the distributive property to multiply x^{3}+15x+1 by 2-2x and combine like terms.

$$6x^{3}-3x^{4}+30x-15x^{2}-\left(2x^{3}-2x^{4}+28x-30x^{2}+2\right)$$

To find the opposite of 2x^{3}-2x^{4}+28x-30x^{2}+2, find the opposite of each term.

$$6x^{3}-3x^{4}+30x-15x^{2}-2x^{3}+2x^{4}-28x+30x^{2}-2$$

Combine 6x^{3} and -2x^{3} to get 4x^{3}.

$$4x^{3}-3x^{4}+30x-15x^{2}+2x^{4}-28x+30x^{2}-2$$

Combine -3x^{4} and 2x^{4} to get -x^{4}.

$$4x^{3}-x^{4}+30x-15x^{2}-28x+30x^{2}-2$$

Combine 30x and -28x to get 2x.

$$4x^{3}-x^{4}+2x-15x^{2}+30x^{2}-2$$

Combine -15x^{2} and 30x^{2} to get 15x^{2}.

$$4x^{3}-x^{4}+2x+15x^{2}-2$$