Solve (4*[(x-1)^3+(x+1)^3*(x+3)]*(x-1)+8):8x-(x^2-2x)*(x^2+2x)

$$( 4 \cdot [ ( x - 1 ) ^ { 3 } + ( x + 1 ) ^ { 3 } \cdot ( x + 3 ) ] \cdot ( x - 1 ) + 8 ) : 8 x - ( x ^ { 2 } - 2 x ) \cdot ( x ^ { 2 } + 2 x )$$

Evaluate

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

Short Solution Steps

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

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

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

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

Use the distributive property to multiply 4\left(x-1\right)^{3}+4\left(x+1\right)^{3}x+12\left(x+1\right)^{3} by x-1 and combine like terms.

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

Express \frac{4\left(x-1\right)^{3}x-4\left(x-1\right)^{3}+4\left(x+1\right)^{3}x^{2}+8x\left(x+1\right)^{3}-12\left(x+1\right)^{3}+8}{8}x as a single fraction.

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

Consider \left(x^{2}-2x\right)\left(x^{2}+2x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

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

Expand \left(2x\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply -x^{4}+4x^{2} times \frac{8}{8}.

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

Since \frac{\left(4\left(x-1\right)^{3}x-4\left(x-1\right)^{3}+4\left(x+1\right)^{3}x^{2}+8x\left(x+1\right)^{3}-12\left(x+1\right)^{3}+8\right)x}{8} and \frac{8\left(-x^{4}+4x^{2}\right)}{8} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(4\left(x-1\right)^{3}x-4\left(x-1\right)^{3}+4\left(x+1\right)^{3}x^{2}+8x\left(x+1\right)^{3}-12\left(x+1\right)^{3}+8\right)x+8\left(-x^{4}+4x^{2}\right).

$$\frac{4x^{5}-12x^{4}+12x^{3}-4x^{2}-4x^{4}+12x^{3}-12x^{2}+4x+4x^{6}+12x^{5}+12x^{4}+4x^{3}+8x^{5}+24x^{4}+24x^{3}+8x^{2}-12x^{4}-36x^{3}-36x^{2}-12x+8x-8x^{4}+32x^{2}}{8}$$

Combine like terms in 4x^{5}-12x^{4}+12x^{3}-4x^{2}-4x^{4}+12x^{3}-12x^{2}+4x+4x^{6}+12x^{5}+12x^{4}+4x^{3}+8x^{5}+24x^{4}+24x^{3}+8x^{2}-12x^{4}-36x^{3}-36x^{2}-12x+8x-8x^{4}+32x^{2}.

$$\frac{24x^{5}+16x^{3}-12x^{2}+4x^{6}}{8}$$

Expand

$$\frac{x^{6}}{2}+3x^{5}+2x^{3}-\frac{3x^{2}}{2}$$

Short Solution Steps

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

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

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

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

Use the distributive property to multiply 4\left(x-1\right)^{3}+4\left(x+1\right)^{3}x+12\left(x+1\right)^{3} by x-1 and combine like terms.

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

Express \frac{4\left(x-1\right)^{3}x-4\left(x-1\right)^{3}+4\left(x+1\right)^{3}x^{2}+8x\left(x+1\right)^{3}-12\left(x+1\right)^{3}+8}{8}x as a single fraction.

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

Consider \left(x^{2}-2x\right)\left(x^{2}+2x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

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

Expand \left(2x\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply -x^{4}+4x^{2} times \frac{8}{8}.

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

Since \frac{\left(4\left(x-1\right)^{3}x-4\left(x-1\right)^{3}+4\left(x+1\right)^{3}x^{2}+8x\left(x+1\right)^{3}-12\left(x+1\right)^{3}+8\right)x}{8} and \frac{8\left(-x^{4}+4x^{2}\right)}{8} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(4\left(x-1\right)^{3}x-4\left(x-1\right)^{3}+4\left(x+1\right)^{3}x^{2}+8x\left(x+1\right)^{3}-12\left(x+1\right)^{3}+8\right)x+8\left(-x^{4}+4x^{2}\right).

$$\frac{4x^{5}-12x^{4}+12x^{3}-4x^{2}-4x^{4}+12x^{3}-12x^{2}+4x+4x^{6}+12x^{5}+12x^{4}+4x^{3}+8x^{5}+24x^{4}+24x^{3}+8x^{2}-12x^{4}-36x^{3}-36x^{2}-12x+8x-8x^{4}+32x^{2}}{8}$$

Combine like terms in 4x^{5}-12x^{4}+12x^{3}-4x^{2}-4x^{4}+12x^{3}-12x^{2}+4x+4x^{6}+12x^{5}+12x^{4}+4x^{3}+8x^{5}+24x^{4}+24x^{3}+8x^{2}-12x^{4}-36x^{3}-36x^{2}-12x+8x-8x^{4}+32x^{2}.

$$\frac{24x^{5}+16x^{3}-12x^{2}+4x^{6}}{8}$$