Solve -x^2(8x+2)+(4x+1)(2x-3)+left(4x+1right)^2

$$- { x }^{ 2 } (8x+2)+(4x+1)(2x-3)+ { \left(4x+1 \right) }^{ 2 }$$

Evaluate

$$-8x^{3}+22x^{2}-2x-2$$

Solution Steps

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

$$8\left(-x^{2}\right)x+2\left(-x^{2}\right)+\left(4x+1\right)\left(2x-3\right)+\left(4x+1\right)^{2}$$

Use the distributive property to multiply 4x+1 by 2x-3 and combine like terms.

$$8\left(-x^{2}\right)x+2\left(-x^{2}\right)+8x^{2}-10x-3+\left(4x+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4x+1\right)^{2}.

$$8\left(-x^{2}\right)x+2\left(-x^{2}\right)+8x^{2}-10x-3+16x^{2}+8x+1$$

Combine 8x^{2} and 16x^{2} to get 24x^{2}.

$$8\left(-x^{2}\right)x+2\left(-x^{2}\right)+24x^{2}-10x-3+8x+1$$

Combine -10x and 8x to get -2x.

$$8\left(-x^{2}\right)x+2\left(-x^{2}\right)+24x^{2}-2x-3+1$$

Add -3 and 1 to get -2.

$$8\left(-x^{2}\right)x+2\left(-x^{2}\right)+24x^{2}-2x-2$$

Multiply 8 and -1 to get -8.

$$-8x^{2}x+2\left(-1\right)x^{2}+24x^{2}-2x-2$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$-8x^{3}+2\left(-1\right)x^{2}+24x^{2}-2x-2$$

Multiply 2 and -1 to get -2.

$$-8x^{3}-2x^{2}+24x^{2}-2x-2$$

Combine -2x^{2} and 24x^{2} to get 22x^{2}.

$$-8x^{3}+22x^{2}-2x-2$$

Expand

$$-8x^{3}+22x^{2}-2x-2$$

Solution Steps

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

$$8\left(-x^{2}\right)x+2\left(-x^{2}\right)+\left(4x+1\right)\left(2x-3\right)+\left(4x+1\right)^{2}$$

Use the distributive property to multiply 4x+1 by 2x-3 and combine like terms.

$$8\left(-x^{2}\right)x+2\left(-x^{2}\right)+8x^{2}-10x-3+\left(4x+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4x+1\right)^{2}.

$$8\left(-x^{2}\right)x+2\left(-x^{2}\right)+8x^{2}-10x-3+16x^{2}+8x+1$$

Combine 8x^{2} and 16x^{2} to get 24x^{2}.

$$8\left(-x^{2}\right)x+2\left(-x^{2}\right)+24x^{2}-10x-3+8x+1$$

Combine -10x and 8x to get -2x.

$$8\left(-x^{2}\right)x+2\left(-x^{2}\right)+24x^{2}-2x-3+1$$

Add -3 and 1 to get -2.

$$8\left(-x^{2}\right)x+2\left(-x^{2}\right)+24x^{2}-2x-2$$

Multiply 8 and -1 to get -8.

$$-8x^{2}x+2\left(-1\right)x^{2}+24x^{2}-2x-2$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$-8x^{3}+2\left(-1\right)x^{2}+24x^{2}-2x-2$$

Multiply 2 and -1 to get -2.

$$-8x^{3}-2x^{2}+24x^{2}-2x-2$$

Combine -2x^{2} and 24x^{2} to get 22x^{2}.

$$-8x^{3}+22x^{2}-2x-2$$