Solve (x^{4}-5x^2-3x)-3/2(3x^3-8x^2-6)-(5x-1)^2

$$( x ^ { 4 } - 5 x ^ { 2 } - 3 x ) - \frac { 3 } { 2 } ( 3 x ^ { 3 } - 8 x ^ { 2 } - 6 ) - ( 5 x - 1 ) ^ { 2 }$$

Evaluate

$$x^{4}-\frac{9x^{3}}{2}-18x^{2}+7x+8$$

Solution Steps

Use the distributive property to multiply -\frac{3}{2} by 3x^{3}-8x^{2}-6.

$$x^{4}-5x^{2}-3x-\frac{9}{2}x^{3}+12x^{2}+9-\left(5x-1\right)^{2}$$

Combine -5x^{2} and 12x^{2} to get 7x^{2}.

$$x^{4}+7x^{2}-3x-\frac{9}{2}x^{3}+9-\left(5x-1\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5x-1\right)^{2}.

$$x^{4}+7x^{2}-3x-\frac{9}{2}x^{3}+9-\left(25x^{2}-10x+1\right)$$

To find the opposite of 25x^{2}-10x+1, find the opposite of each term.

$$x^{4}+7x^{2}-3x-\frac{9}{2}x^{3}+9-25x^{2}+10x-1$$

Combine 7x^{2} and -25x^{2} to get -18x^{2}.

$$x^{4}-18x^{2}-3x-\frac{9}{2}x^{3}+9+10x-1$$

Combine -3x and 10x to get 7x.

$$x^{4}-18x^{2}+7x-\frac{9}{2}x^{3}+9-1$$

Subtract 1 from 9 to get 8.

$$x^{4}-18x^{2}+7x-\frac{9}{2}x^{3}+8$$

Expand

$$x^{4}-\frac{9x^{3}}{2}-18x^{2}+7x+8$$

Solution Steps

Use the distributive property to multiply -\frac{3}{2} by 3x^{3}-8x^{2}-6.

$$x^{4}-5x^{2}-3x-\frac{9}{2}x^{3}+12x^{2}+9-\left(5x-1\right)^{2}$$

Combine -5x^{2} and 12x^{2} to get 7x^{2}.

$$x^{4}+7x^{2}-3x-\frac{9}{2}x^{3}+9-\left(5x-1\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5x-1\right)^{2}.

$$x^{4}+7x^{2}-3x-\frac{9}{2}x^{3}+9-\left(25x^{2}-10x+1\right)$$

To find the opposite of 25x^{2}-10x+1, find the opposite of each term.

$$x^{4}+7x^{2}-3x-\frac{9}{2}x^{3}+9-25x^{2}+10x-1$$

Combine 7x^{2} and -25x^{2} to get -18x^{2}.

$$x^{4}-18x^{2}-3x-\frac{9}{2}x^{3}+9+10x-1$$

Combine -3x and 10x to get 7x.

$$x^{4}-18x^{2}+7x-\frac{9}{2}x^{3}+9-1$$

Subtract 1 from 9 to get 8.

$$x^{4}-18x^{2}+7x-\frac{9}{2}x^{3}+8$$