Solve (+1/2x^{2}-1)(+18x^{4}-1)-(-3x^3)^2-9/2x(-4x^3-x)-(+4x^2+1)

$$( + \frac { 1 } { 2 } x ^ { 2 } - 1 ) ( + 18 x ^ { 4 } - 1 ) - ( - 3 x ^ { 3 } ) ^ { 2 } - \frac { 9 } { 2 } x ( - 4 x ^ { 3 } - x ) - ( + 4 x ^ { 2 } + 1 )$$

Evaluate

$$0$$

Solution Steps

Use the distributive property to multiply \frac{1}{2}x^{2}-1 by 18x^{4}-1.

$$9x^{6}-\frac{1}{2}x^{2}-18x^{4}+1-\left(-3x^{3}\right)^{2}-\frac{9}{2}x\left(-4x^{3}-x\right)-\left(4x^{2}+1\right)$$

Expand \left(-3x^{3}\right)^{2}.

$$9x^{6}-\frac{1}{2}x^{2}-18x^{4}+1-\left(-3\right)^{2}\left(x^{3}\right)^{2}-\frac{9}{2}x\left(-4x^{3}-x\right)-\left(4x^{2}+1\right)$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$9x^{6}-\frac{1}{2}x^{2}-18x^{4}+1-\left(-3\right)^{2}x^{6}-\frac{9}{2}x\left(-4x^{3}-x\right)-\left(4x^{2}+1\right)$$

Calculate -3 to the power of 2 and get 9.

$$9x^{6}-\frac{1}{2}x^{2}-18x^{4}+1-9x^{6}-\frac{9}{2}x\left(-4x^{3}-x\right)-\left(4x^{2}+1\right)$$

Combine 9x^{6} and -9x^{6} to get 0.

$$-\frac{1}{2}x^{2}-18x^{4}+1-\frac{9}{2}x\left(-4x^{3}-x\right)-\left(4x^{2}+1\right)$$

To find the opposite of 4x^{2}+1, find the opposite of each term.

$$-\frac{1}{2}x^{2}-18x^{4}+1-\frac{9}{2}x\left(-4x^{3}-x\right)-4x^{2}-1$$

Use the distributive property to multiply -\frac{9}{2}x by -4x^{3}-x.

$$-\frac{1}{2}x^{2}-18x^{4}+1+18x^{4}+\frac{9}{2}x^{2}-4x^{2}-1$$

Combine -18x^{4} and 18x^{4} to get 0.

$$-\frac{1}{2}x^{2}+1+\frac{9}{2}x^{2}-4x^{2}-1$$

Combine -\frac{1}{2}x^{2} and \frac{9}{2}x^{2} to get 4x^{2}.

$$4x^{2}+1-4x^{2}-1$$

Combine 4x^{2} and -4x^{2} to get 0.

$$1-1$$

Subtract 1 from 1 to get 0.

$$0$$

Factor

$$0$$

Solution Steps

Factor out \frac{1}{2}.

$$\frac{\left(x^{2}-2\right)\left(18x^{4}-1\right)-18x^{6}-9x\left(-4x^{3}-x\right)-2\left(4x^{2}+1\right)}{2}$$

Simplify.

$$0$$