Solve (-x+2)(x-1)quadtext{(4)}(3x^4-3x^2+1)(x^4+x^2-2)

$$( - x + 2 ) ( x - 1 ) \quad \text { (4) } ( 3 x ^ { 4 } - 3 x ^ { 2 } + 1 ) ( x ^ { 4 } + x ^ { 2 } - 2 )$$

Evaluate

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

Solution Steps

Use the distributive property to multiply -x+2 by x-1.

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

Multiply -1 and -1 to get 1.

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

Combine x and 2x to get 3x.

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

Use the distributive property to multiply \left(-x\right)x+3x-2 by 4.

$$\left(4\left(-x\right)x+12x-8\right)\left(3x^{4}-3x^{2}+1\right)\left(x^{4}+x^{2}-2\right)$$

Use the distributive property to multiply 4\left(-x\right)x+12x-8 by 3x^{4}-3x^{2}+1.

$$\left(12\left(-x\right)x^{5}-12\left(-x\right)x^{3}+4\left(-x\right)x+36x^{5}-36x^{3}+12x-24x^{4}+24x^{2}-8\right)\left(x^{4}+x^{2}-2\right)$$

Multiply -12 and -1 to get 12.

$$\left(12\left(-x\right)x^{5}+12xx^{3}+4\left(-x\right)x+36x^{5}-36x^{3}+12x-24x^{4}+24x^{2}-8\right)\left(x^{4}+x^{2}-2\right)$$

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

$$\left(12\left(-x\right)x^{5}+12x^{4}+4\left(-x\right)x+36x^{5}-36x^{3}+12x-24x^{4}+24x^{2}-8\right)\left(x^{4}+x^{2}-2\right)$$

Combine 12x^{4} and -24x^{4} to get -12x^{4}.

$$\left(12\left(-x\right)x^{5}-12x^{4}+4\left(-x\right)x+36x^{5}-36x^{3}+12x+24x^{2}-8\right)\left(x^{4}+x^{2}-2\right)$$

Use the distributive property to multiply 12\left(-x\right)x^{5}-12x^{4}+4\left(-x\right)x+36x^{5}-36x^{3}+12x+24x^{2}-8 by x^{4}+x^{2}-2 and combine like terms.

$$12\left(-x\right)x^{9}+12\left(-x\right)x^{7}-20\left(-x\right)x^{5}-12x^{8}+12x^{6}+40x^{4}+4\left(-x\right)x^{3}-8\left(-x\right)x+36x^{9}-96x^{5}+84x^{3}-24x-56x^{2}+16$$

Multiply -20 and -1 to get 20.

$$12\left(-x\right)x^{9}+12\left(-x\right)x^{7}+20xx^{5}-12x^{8}+12x^{6}+40x^{4}+4\left(-x\right)x^{3}-8\left(-x\right)x+36x^{9}-96x^{5}+84x^{3}-24x-56x^{2}+16$$

To multiply powers of the same base, add their exponents. Add 1 and 5 to get 6.

$$12\left(-x\right)x^{9}+12\left(-x\right)x^{7}+20x^{6}-12x^{8}+12x^{6}+40x^{4}+4\left(-x\right)x^{3}-8\left(-x\right)x+36x^{9}-96x^{5}+84x^{3}-24x-56x^{2}+16$$

Combine 20x^{6} and 12x^{6} to get 32x^{6}.

$$12\left(-x\right)x^{9}+12\left(-x\right)x^{7}+32x^{6}-12x^{8}+40x^{4}+4\left(-x\right)x^{3}-8\left(-x\right)x+36x^{9}-96x^{5}+84x^{3}-24x-56x^{2}+16$$

Multiply -8 and -1 to get 8.

$$12\left(-x\right)x^{9}+12\left(-x\right)x^{7}+32x^{6}-12x^{8}+40x^{4}+4\left(-x\right)x^{3}+8xx+36x^{9}-96x^{5}+84x^{3}-24x-56x^{2}+16$$

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

$$12\left(-x\right)x^{9}+12\left(-x\right)x^{7}+32x^{6}-12x^{8}+40x^{4}+4\left(-x\right)x^{3}+8x^{2}+36x^{9}-96x^{5}+84x^{3}-24x-56x^{2}+16$$

Combine 8x^{2} and -56x^{2} to get -48x^{2}.

$$12\left(-x\right)x^{9}+12\left(-x\right)x^{7}+32x^{6}-12x^{8}+40x^{4}+4\left(-x\right)x^{3}-48x^{2}+36x^{9}-96x^{5}+84x^{3}-24x+16$$

Multiply 12 and -1 to get -12.

$$-12xx^{9}+12\left(-1\right)xx^{7}+32x^{6}-12x^{8}+40x^{4}+4\left(-1\right)xx^{3}-48x^{2}+36x^{9}-96x^{5}+84x^{3}-24x+16$$

To multiply powers of the same base, add their exponents. Add 1 and 9 to get 10.

$$-12x^{10}+12\left(-1\right)xx^{7}+32x^{6}-12x^{8}+40x^{4}+4\left(-1\right)xx^{3}-48x^{2}+36x^{9}-96x^{5}+84x^{3}-24x+16$$

To multiply powers of the same base, add their exponents. Add 1 and 7 to get 8.

$$-12x^{10}+12\left(-1\right)x^{8}+32x^{6}-12x^{8}+40x^{4}+4\left(-1\right)xx^{3}-48x^{2}+36x^{9}-96x^{5}+84x^{3}-24x+16$$

Multiply 12 and -1 to get -12.

$$-12x^{10}-12x^{8}+32x^{6}-12x^{8}+40x^{4}+4\left(-1\right)xx^{3}-48x^{2}+36x^{9}-96x^{5}+84x^{3}-24x+16$$

Combine -12x^{8} and -12x^{8} to get -24x^{8}.

$$-12x^{10}-24x^{8}+32x^{6}+40x^{4}+4\left(-1\right)xx^{3}-48x^{2}+36x^{9}-96x^{5}+84x^{3}-24x+16$$

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

$$-12x^{10}-24x^{8}+32x^{6}+40x^{4}+4\left(-1\right)x^{4}-48x^{2}+36x^{9}-96x^{5}+84x^{3}-24x+16$$

Multiply 4 and -1 to get -4.

$$-12x^{10}-24x^{8}+32x^{6}+40x^{4}-4x^{4}-48x^{2}+36x^{9}-96x^{5}+84x^{3}-24x+16$$

Combine 40x^{4} and -4x^{4} to get 36x^{4}.

$$-12x^{10}-24x^{8}+32x^{6}+36x^{4}-48x^{2}+36x^{9}-96x^{5}+84x^{3}-24x+16$$

Expand

$$16-24x-48x^{2}+84x^{3}+36x^{4}-96x^{5}+32x^{6}-24x^{8}+36x^{9}-12x^{10}$$

Solution Steps

Use the distributive property to multiply -x+2 by x-1.

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

Multiply -1 and -1 to get 1.

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

Combine x and 2x to get 3x.

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

Use the distributive property to multiply \left(-x\right)x+3x-2 by 4.

$$\left(4\left(-x\right)x+12x-8\right)\left(3x^{4}-3x^{2}+1\right)\left(x^{4}+x^{2}-2\right)$$

Use the distributive property to multiply 4\left(-x\right)x+12x-8 by 3x^{4}-3x^{2}+1.

$$\left(12\left(-x\right)x^{5}-12\left(-x\right)x^{3}+4\left(-x\right)x+36x^{5}-36x^{3}+12x-24x^{4}+24x^{2}-8\right)\left(x^{4}+x^{2}-2\right)$$

Multiply -12 and -1 to get 12.

$$\left(12\left(-x\right)x^{5}+12xx^{3}+4\left(-x\right)x+36x^{5}-36x^{3}+12x-24x^{4}+24x^{2}-8\right)\left(x^{4}+x^{2}-2\right)$$

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

$$\left(12\left(-x\right)x^{5}+12x^{4}+4\left(-x\right)x+36x^{5}-36x^{3}+12x-24x^{4}+24x^{2}-8\right)\left(x^{4}+x^{2}-2\right)$$

Combine 12x^{4} and -24x^{4} to get -12x^{4}.

$$\left(12\left(-x\right)x^{5}-12x^{4}+4\left(-x\right)x+36x^{5}-36x^{3}+12x+24x^{2}-8\right)\left(x^{4}+x^{2}-2\right)$$

Use the distributive property to multiply 12\left(-x\right)x^{5}-12x^{4}+4\left(-x\right)x+36x^{5}-36x^{3}+12x+24x^{2}-8 by x^{4}+x^{2}-2 and combine like terms.

$$12\left(-x\right)x^{9}+12\left(-x\right)x^{7}-20\left(-x\right)x^{5}-12x^{8}+12x^{6}+40x^{4}+4\left(-x\right)x^{3}-8\left(-x\right)x+36x^{9}-96x^{5}+84x^{3}-24x-56x^{2}+16$$

Multiply -20 and -1 to get 20.

$$12\left(-x\right)x^{9}+12\left(-x\right)x^{7}+20xx^{5}-12x^{8}+12x^{6}+40x^{4}+4\left(-x\right)x^{3}-8\left(-x\right)x+36x^{9}-96x^{5}+84x^{3}-24x-56x^{2}+16$$

To multiply powers of the same base, add their exponents. Add 1 and 5 to get 6.

$$12\left(-x\right)x^{9}+12\left(-x\right)x^{7}+20x^{6}-12x^{8}+12x^{6}+40x^{4}+4\left(-x\right)x^{3}-8\left(-x\right)x+36x^{9}-96x^{5}+84x^{3}-24x-56x^{2}+16$$

Combine 20x^{6} and 12x^{6} to get 32x^{6}.

$$12\left(-x\right)x^{9}+12\left(-x\right)x^{7}+32x^{6}-12x^{8}+40x^{4}+4\left(-x\right)x^{3}-8\left(-x\right)x+36x^{9}-96x^{5}+84x^{3}-24x-56x^{2}+16$$

Multiply -8 and -1 to get 8.

$$12\left(-x\right)x^{9}+12\left(-x\right)x^{7}+32x^{6}-12x^{8}+40x^{4}+4\left(-x\right)x^{3}+8xx+36x^{9}-96x^{5}+84x^{3}-24x-56x^{2}+16$$

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

$$12\left(-x\right)x^{9}+12\left(-x\right)x^{7}+32x^{6}-12x^{8}+40x^{4}+4\left(-x\right)x^{3}+8x^{2}+36x^{9}-96x^{5}+84x^{3}-24x-56x^{2}+16$$

Combine 8x^{2} and -56x^{2} to get -48x^{2}.

$$12\left(-x\right)x^{9}+12\left(-x\right)x^{7}+32x^{6}-12x^{8}+40x^{4}+4\left(-x\right)x^{3}-48x^{2}+36x^{9}-96x^{5}+84x^{3}-24x+16$$

Multiply 12 and -1 to get -12.

$$-12xx^{9}+12\left(-1\right)xx^{7}+32x^{6}-12x^{8}+40x^{4}+4\left(-1\right)xx^{3}-48x^{2}+36x^{9}-96x^{5}+84x^{3}-24x+16$$

To multiply powers of the same base, add their exponents. Add 1 and 9 to get 10.

$$-12x^{10}+12\left(-1\right)xx^{7}+32x^{6}-12x^{8}+40x^{4}+4\left(-1\right)xx^{3}-48x^{2}+36x^{9}-96x^{5}+84x^{3}-24x+16$$

To multiply powers of the same base, add their exponents. Add 1 and 7 to get 8.

$$-12x^{10}+12\left(-1\right)x^{8}+32x^{6}-12x^{8}+40x^{4}+4\left(-1\right)xx^{3}-48x^{2}+36x^{9}-96x^{5}+84x^{3}-24x+16$$

Multiply 12 and -1 to get -12.

$$-12x^{10}-12x^{8}+32x^{6}-12x^{8}+40x^{4}+4\left(-1\right)xx^{3}-48x^{2}+36x^{9}-96x^{5}+84x^{3}-24x+16$$

Combine -12x^{8} and -12x^{8} to get -24x^{8}.

$$-12x^{10}-24x^{8}+32x^{6}+40x^{4}+4\left(-1\right)xx^{3}-48x^{2}+36x^{9}-96x^{5}+84x^{3}-24x+16$$

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

$$-12x^{10}-24x^{8}+32x^{6}+40x^{4}+4\left(-1\right)x^{4}-48x^{2}+36x^{9}-96x^{5}+84x^{3}-24x+16$$

Multiply 4 and -1 to get -4.

$$-12x^{10}-24x^{8}+32x^{6}+40x^{4}-4x^{4}-48x^{2}+36x^{9}-96x^{5}+84x^{3}-24x+16$$

Combine 40x^{4} and -4x^{4} to get 36x^{4}.

$$-12x^{10}-24x^{8}+32x^{6}+36x^{4}-48x^{2}+36x^{9}-96x^{5}+84x^{3}-24x+16$$