Solve x(15-x(2x+5)(x-3)+x(-15-7x(x-4)7x(x-4

$$x(15-x(2x+5)(x-3)+x(-15-7x(x-4)7x(x-4$$

Evaluate

$$x\left(-49\left(x-4\right)^{2}x^{3}-2x^{3}+x^{2}+15\right)$$

Solution Steps

Multiply x and x to get x^{2}.

$$x\left(15-x\left(2x+5\right)\left(x-3\right)+x\left(-15-7x^{2}\left(x-4\right)\times 7\left(x-4\right)\right)\right)$$

Multiply x-4 and x-4 to get \left(x-4\right)^{2}.

$$x\left(15-x\left(2x+5\right)\left(x-3\right)+x\left(-15-7x^{2}\left(x-4\right)^{2}\times 7\right)\right)$$

Use the distributive property to multiply x by 2x+5.

$$x\left(15-\left(2x^{2}+5x\right)\left(x-3\right)+x\left(-15-7x^{2}\left(x-4\right)^{2}\times 7\right)\right)$$

Apply the distributive property by multiplying each term of 2x^{2}+5x by each term of x-3.

$$x\left(15-\left(2x^{3}-6x^{2}+5x^{2}-15x\right)+x\left(-15-7x^{2}\left(x-4\right)^{2}\times 7\right)\right)$$

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

$$x\left(15-\left(2x^{3}-x^{2}-15x\right)+x\left(-15-7x^{2}\left(x-4\right)^{2}\times 7\right)\right)$$

To find the opposite of 2x^{3}-x^{2}-15x, find the opposite of each term.

$$x\left(15-2x^{3}-\left(-x^{2}\right)-\left(-15x\right)+x\left(-15-7x^{2}\left(x-4\right)^{2}\times 7\right)\right)$$

The opposite of -x^{2} is x^{2}.

$$x\left(15-2x^{3}+x^{2}-\left(-15x\right)+x\left(-15-7x^{2}\left(x-4\right)^{2}\times 7\right)\right)$$

The opposite of -15x is 15x.

$$x\left(15-2x^{3}+x^{2}+15x+x\left(-15-7x^{2}\left(x-4\right)^{2}\times 7\right)\right)$$

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

$$x\left(15-2x^{3}+x^{2}+15x+x\left(-15-7x^{2}\left(x^{2}-8x+16\right)\times 7\right)\right)$$

Multiply 7 and 7 to get 49.

$$x\left(15-2x^{3}+x^{2}+15x+x\left(-15-49x^{2}\left(x^{2}-8x+16\right)\right)\right)$$

Use the distributive property to multiply x by 15-2x^{3}+x^{2}+15x+x\left(-15-49x^{2}\left(x^{2}-8x+16\right)\right).

$$15x-2x^{4}+x^{3}+15x^{2}+\left(-15-49x^{2}\left(x^{2}-8x+16\right)\right)x^{2}$$

Use the distributive property to multiply -49x^{2} by x^{2}-8x+16.

$$15x-2x^{4}+x^{3}+15x^{2}+\left(-15-49x^{4}+392x^{3}-784x^{2}\right)x^{2}$$

Use the distributive property to multiply -15-49x^{4}+392x^{3}-784x^{2} by x^{2}.

$$15x-2x^{4}+x^{3}+15x^{2}-15x^{2}-49x^{6}+392x^{5}-784x^{4}$$

Combine 15x^{2} and -15x^{2} to get 0.

$$15x-2x^{4}+x^{3}-49x^{6}+392x^{5}-784x^{4}$$

Combine -2x^{4} and -784x^{4} to get -786x^{4}.

$$15x-786x^{4}+x^{3}-49x^{6}+392x^{5}$$

Expand

$$15x+x^{3}-786x^{4}+392x^{5}-49x^{6}$$

Solution Steps

Multiply x and x to get x^{2}.

$$x\left(15-x\left(2x+5\right)\left(x-3\right)+x\left(-15-7x^{2}\left(x-4\right)\times 7\left(x-4\right)\right)\right)$$

Multiply x-4 and x-4 to get \left(x-4\right)^{2}.

$$x\left(15-x\left(2x+5\right)\left(x-3\right)+x\left(-15-7x^{2}\left(x-4\right)^{2}\times 7\right)\right)$$

Use the distributive property to multiply x by 2x+5.

$$x\left(15-\left(2x^{2}+5x\right)\left(x-3\right)+x\left(-15-7x^{2}\left(x-4\right)^{2}\times 7\right)\right)$$

Apply the distributive property by multiplying each term of 2x^{2}+5x by each term of x-3.

$$x\left(15-\left(2x^{3}-6x^{2}+5x^{2}-15x\right)+x\left(-15-7x^{2}\left(x-4\right)^{2}\times 7\right)\right)$$

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

$$x\left(15-\left(2x^{3}-x^{2}-15x\right)+x\left(-15-7x^{2}\left(x-4\right)^{2}\times 7\right)\right)$$

To find the opposite of 2x^{3}-x^{2}-15x, find the opposite of each term.

$$x\left(15-2x^{3}-\left(-x^{2}\right)-\left(-15x\right)+x\left(-15-7x^{2}\left(x-4\right)^{2}\times 7\right)\right)$$

The opposite of -x^{2} is x^{2}.

$$x\left(15-2x^{3}+x^{2}-\left(-15x\right)+x\left(-15-7x^{2}\left(x-4\right)^{2}\times 7\right)\right)$$

The opposite of -15x is 15x.

$$x\left(15-2x^{3}+x^{2}+15x+x\left(-15-7x^{2}\left(x-4\right)^{2}\times 7\right)\right)$$

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

$$x\left(15-2x^{3}+x^{2}+15x+x\left(-15-7x^{2}\left(x^{2}-8x+16\right)\times 7\right)\right)$$

Multiply 7 and 7 to get 49.

$$x\left(15-2x^{3}+x^{2}+15x+x\left(-15-49x^{2}\left(x^{2}-8x+16\right)\right)\right)$$

Use the distributive property to multiply x by 15-2x^{3}+x^{2}+15x+x\left(-15-49x^{2}\left(x^{2}-8x+16\right)\right).

$$15x-2x^{4}+x^{3}+15x^{2}+\left(-15-49x^{2}\left(x^{2}-8x+16\right)\right)x^{2}$$

Use the distributive property to multiply -49x^{2} by x^{2}-8x+16.

$$15x-2x^{4}+x^{3}+15x^{2}+\left(-15-49x^{4}+392x^{3}-784x^{2}\right)x^{2}$$

Use the distributive property to multiply -15-49x^{4}+392x^{3}-784x^{2} by x^{2}.

$$15x-2x^{4}+x^{3}+15x^{2}-15x^{2}-49x^{6}+392x^{5}-784x^{4}$$

Combine 15x^{2} and -15x^{2} to get 0.

$$15x-2x^{4}+x^{3}-49x^{6}+392x^{5}-784x^{4}$$

Combine -2x^{4} and -784x^{4} to get -786x^{4}.

$$15x-786x^{4}+x^{3}-49x^{6}+392x^{5}$$