Solve (2x^2+7x+2)+(x+2)

$$( 2 x ^ { 2 } + 7 x + 2 ) + ( x + 2 )$$

Evaluate

$$2\left(x^{2}+4x+2\right)$$

Solution Steps

Combine 7x and x to get 8x.

$$2x^{2}+8x+2+2$$

Add 2 and 2 to get 4.

$$2x^{2}+8x+4$$

Factor

$$2\left(x-\left(-\sqrt{2}-2\right)\right)\left(x-\left(\sqrt{2}-2\right)\right)$$

Steps Using the Quadratic Formula

Combine 7x and x to get 8x.

$$factor(2x^{2}+8x+2+2)$$

Add 2 and 2 to get 4.

$$factor(2x^{2}+8x+4)$$

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.

$$2x^{2}+8x+4=0$$

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.

$$x=\frac{-8±\sqrt{8^{2}-4\times 2\times 4}}{2\times 2}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\times 2\times 4}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-8±\sqrt{64-8\times 4}}{2\times 2}$$

Multiply -8 times 4.

$$x=\frac{-8±\sqrt{64-32}}{2\times 2}$$

Add 64 to -32.

$$x=\frac{-8±\sqrt{32}}{2\times 2}$$

Take the square root of 32.

$$x=\frac{-8±4\sqrt{2}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-8±4\sqrt{2}}{4}$$

Now solve the equation x=\frac{-8±4\sqrt{2}}{4} when ± is plus. Add -8 to 4\sqrt{2}.

$$x=\frac{4\sqrt{2}-8}{4}$$

Divide -8+4\sqrt{2} by 4.

$$x=\sqrt{2}-2$$

Now solve the equation x=\frac{-8±4\sqrt{2}}{4} when ± is minus. Subtract 4\sqrt{2} from -8.

$$x=\frac{-4\sqrt{2}-8}{4}$$

Divide -8-4\sqrt{2} by 4.

$$x=-\sqrt{2}-2$$

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

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