Solve -16x+(x-2)(x^2-4)-3x=(x+2)(x^2+4x+4)

$$- 16 x + ( x - 2 ) ( x ^ { 2 } - 4 ) - 3 x = ( x + 2 ) ( x ^ { 2 } + 4 x + 4 )$$

Solve for x

$$x = -\frac{35}{8} = -4\frac{3}{8} = -4.375$$

$$x=0$$

Steps Using Factoring

Use the distributive property to multiply x-2 by x^{2}-4.

$$-16x+x^{3}-4x-2x^{2}+8-3x=\left(x+2\right)\left(x^{2}+4x+4\right)$$

Combine -16x and -4x to get -20x.

$$-20x+x^{3}-2x^{2}+8-3x=\left(x+2\right)\left(x^{2}+4x+4\right)$$

Combine -20x and -3x to get -23x.

$$-23x+x^{3}-2x^{2}+8=\left(x+2\right)\left(x^{2}+4x+4\right)$$

Use the distributive property to multiply x+2 by x^{2}+4x+4 and combine like terms.

$$-23x+x^{3}-2x^{2}+8=x^{3}+6x^{2}+12x+8$$

Subtract x^{3} from both sides.

$$-23x+x^{3}-2x^{2}+8-x^{3}=6x^{2}+12x+8$$

Combine x^{3} and -x^{3} to get 0.

$$-23x-2x^{2}+8=6x^{2}+12x+8$$

Subtract 6x^{2} from both sides.

$$-23x-2x^{2}+8-6x^{2}=12x+8$$

Combine -2x^{2} and -6x^{2} to get -8x^{2}.

$$-23x-8x^{2}+8=12x+8$$

Subtract 12x from both sides.

$$-23x-8x^{2}+8-12x=8$$

Combine -23x and -12x to get -35x.

$$-35x-8x^{2}+8=8$$

Subtract 8 from both sides.

$$-35x-8x^{2}+8-8=0$$

Subtract 8 from 8 to get 0.

$$-35x-8x^{2}=0$$

Factor out x.

$$x\left(-35-8x\right)=0$$

To find equation solutions, solve x=0 and -35-8x=0.

$$x=0 x=-\frac{35}{8}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-2 by x^{2}-4.

$$-16x+x^{3}-4x-2x^{2}+8-3x=\left(x+2\right)\left(x^{2}+4x+4\right)$$

Combine -16x and -4x to get -20x.

$$-20x+x^{3}-2x^{2}+8-3x=\left(x+2\right)\left(x^{2}+4x+4\right)$$

Combine -20x and -3x to get -23x.

$$-23x+x^{3}-2x^{2}+8=\left(x+2\right)\left(x^{2}+4x+4\right)$$

Use the distributive property to multiply x+2 by x^{2}+4x+4 and combine like terms.

$$-23x+x^{3}-2x^{2}+8=x^{3}+6x^{2}+12x+8$$

Subtract x^{3} from both sides.

$$-23x+x^{3}-2x^{2}+8-x^{3}=6x^{2}+12x+8$$

Combine x^{3} and -x^{3} to get 0.

$$-23x-2x^{2}+8=6x^{2}+12x+8$$

Subtract 6x^{2} from both sides.

$$-23x-2x^{2}+8-6x^{2}=12x+8$$

Combine -2x^{2} and -6x^{2} to get -8x^{2}.

$$-23x-8x^{2}+8=12x+8$$

Subtract 12x from both sides.

$$-23x-8x^{2}+8-12x=8$$

Combine -23x and -12x to get -35x.

$$-35x-8x^{2}+8=8$$

Subtract 8 from both sides.

$$-35x-8x^{2}+8-8=0$$

Subtract 8 from 8 to get 0.

$$-35x-8x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-8x^{2}-35x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, -35 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-35\right)±\sqrt{\left(-35\right)^{2}}}{2\left(-8\right)}$$

Take the square root of \left(-35\right)^{2}.

$$x=\frac{-\left(-35\right)±35}{2\left(-8\right)}$$

The opposite of -35 is 35.

$$x=\frac{35±35}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{35±35}{-16}$$

Now solve the equation x=\frac{35±35}{-16} when ± is plus. Add 35 to 35.

$$x=\frac{70}{-16}$$

Reduce the fraction \frac{70}{-16} to lowest terms by extracting and canceling out 2.

$$x=-\frac{35}{8}$$

Now solve the equation x=\frac{35±35}{-16} when ± is minus. Subtract 35 from 35.

$$x=\frac{0}{-16}$$

Divide 0 by -16.

$$x=0$$

The equation is now solved.

$$x=-\frac{35}{8} x=0$$

Steps for Completing the Square

Use the distributive property to multiply x-2 by x^{2}-4.

$$-16x+x^{3}-4x-2x^{2}+8-3x=\left(x+2\right)\left(x^{2}+4x+4\right)$$

Combine -16x and -4x to get -20x.

$$-20x+x^{3}-2x^{2}+8-3x=\left(x+2\right)\left(x^{2}+4x+4\right)$$

Combine -20x and -3x to get -23x.

$$-23x+x^{3}-2x^{2}+8=\left(x+2\right)\left(x^{2}+4x+4\right)$$

Use the distributive property to multiply x+2 by x^{2}+4x+4 and combine like terms.

$$-23x+x^{3}-2x^{2}+8=x^{3}+6x^{2}+12x+8$$

Subtract x^{3} from both sides.

$$-23x+x^{3}-2x^{2}+8-x^{3}=6x^{2}+12x+8$$

Combine x^{3} and -x^{3} to get 0.

$$-23x-2x^{2}+8=6x^{2}+12x+8$$

Subtract 6x^{2} from both sides.

$$-23x-2x^{2}+8-6x^{2}=12x+8$$

Combine -2x^{2} and -6x^{2} to get -8x^{2}.

$$-23x-8x^{2}+8=12x+8$$

Subtract 12x from both sides.

$$-23x-8x^{2}+8-12x=8$$

Combine -23x and -12x to get -35x.

$$-35x-8x^{2}+8=8$$

Subtract 8 from both sides.

$$-35x-8x^{2}=8-8$$

Subtract 8 from 8 to get 0.

$$-35x-8x^{2}=0$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-8x^{2}-35x=0$$

Divide both sides by -8.

$$\frac{-8x^{2}-35x}{-8}=\frac{0}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x^{2}+\left(-\frac{35}{-8}\right)x=\frac{0}{-8}$$

Divide -35 by -8.

$$x^{2}+\frac{35}{8}x=\frac{0}{-8}$$

Divide 0 by -8.

$$x^{2}+\frac{35}{8}x=0$$

Divide \frac{35}{8}, the coefficient of the x term, by 2 to get \frac{35}{16}. Then add the square of \frac{35}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{35}{8}x+\left(\frac{35}{16}\right)^{2}=\left(\frac{35}{16}\right)^{2}$$

Square \frac{35}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{35}{8}x+\frac{1225}{256}=\frac{1225}{256}$$

Factor x^{2}+\frac{35}{8}x+\frac{1225}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{35}{16}\right)^{2}=\frac{1225}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{35}{16}\right)^{2}}=\sqrt{\frac{1225}{256}}$$

Simplify.

$$x+\frac{35}{16}=\frac{35}{16} x+\frac{35}{16}=-\frac{35}{16}$$

Subtract \frac{35}{16} from both sides of the equation.

$$x=0 x=-\frac{35}{8}$$