Solve (2-8x-4x^2)+(-x^2+3x+2)

$$( 2 - 8 x - 4 x ^ { 2 } ) + ( - x ^ { 2 } + 3 x + 2 )$$

Evaluate

$$4-5x-5x^{2}$$

Solution Steps

Combine -8x and 3x to get -5x.

$$2-5x-4x^{2}-x^{2}+2$$

Add 2 and 2 to get 4.

$$4-5x-4x^{2}-x^{2}$$

Combine -4x^{2} and -x^{2} to get -5x^{2}.

$$4-5x-5x^{2}$$

Factor

$$-5\left(x-\left(-\frac{\sqrt{105}}{10}-\frac{1}{2}\right)\right)\left(x-\left(\frac{\sqrt{105}}{10}-\frac{1}{2}\right)\right)$$

Steps Using the Quadratic Formula

Combine -8x and 3x to get -5x.

$$factor(2-5x-4x^{2}-x^{2}+2)$$

Add 2 and 2 to get 4.

$$factor(4-5x-4x^{2}-x^{2})$$

Combine -4x^{2} and -x^{2} to get -5x^{2}.

$$factor(4-5x-5x^{2})$$

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-5x^{2}-5x+4=0$$

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.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-5\right)\times 4}}{2\left(-5\right)}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\left(-5\right)\times 4}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-\left(-5\right)±\sqrt{25+20\times 4}}{2\left(-5\right)}$$

Multiply 20 times 4.

$$x=\frac{-\left(-5\right)±\sqrt{25+80}}{2\left(-5\right)}$$

Add 25 to 80.

$$x=\frac{-\left(-5\right)±\sqrt{105}}{2\left(-5\right)}$$

The opposite of -5 is 5.

$$x=\frac{5±\sqrt{105}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{5±\sqrt{105}}{-10}$$

Now solve the equation x=\frac{5±\sqrt{105}}{-10} when ± is plus. Add 5 to \sqrt{105}.

$$x=\frac{\sqrt{105}+5}{-10}$$

Divide 5+\sqrt{105} by -10.

$$x=-\frac{\sqrt{105}}{10}-\frac{1}{2}$$

Now solve the equation x=\frac{5±\sqrt{105}}{-10} when ± is minus. Subtract \sqrt{105} from 5.

$$x=\frac{5-\sqrt{105}}{-10}$$

Divide 5-\sqrt{105} by -10.

$$x=\frac{\sqrt{105}}{10}-\frac{1}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{1}{2}-\frac{\sqrt{105}}{10} for x_{1} and -\frac{1}{2}+\frac{\sqrt{105}}{10} for x_{2}.

$$-5x^{2}-5x+4=-5\left(x-\left(-\frac{\sqrt{105}}{10}-\frac{1}{2}\right)\right)\left(x-\left(\frac{\sqrt{105}}{10}-\frac{1}{2}\right)\right)$$