Solve 6left(x+1right)^2+2(x-1)(x^2+x+1)-2left(x-1right)^3=32

$$6 { \left(x+1 \right) }^{ 2 } +2(x-1)( { x }^{ 2 } +x+1)-2 { \left(x-1 \right) }^{ 3 } =32$$

Solve for x

$$x=\frac{\sqrt{321}}{12}-\frac{1}{4}\approx 1.243039406$$

$$x=-\frac{\sqrt{321}}{12}-\frac{1}{4}\approx -1.743039406$$

Steps Using the Quadratic Formula

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

$$6\left(x^{2}+2x+1\right)+2\left(x-1\right)\left(x^{2}+x+1\right)-2\left(x-1\right)^{3}=32$$

Use the distributive property to multiply 6 by x^{2}+2x+1.

$$6x^{2}+12x+6+2\left(x-1\right)\left(x^{2}+x+1\right)-2\left(x-1\right)^{3}=32$$

Use the distributive property to multiply 2 by x-1.

$$6x^{2}+12x+6+\left(2x-2\right)\left(x^{2}+x+1\right)-2\left(x-1\right)^{3}=32$$

Use the distributive property to multiply 2x-2 by x^{2}+x+1 and combine like terms.

$$6x^{2}+12x+6+2x^{3}-2-2\left(x-1\right)^{3}=32$$

Subtract 2 from 6 to get 4.

$$6x^{2}+12x+4+2x^{3}-2\left(x-1\right)^{3}=32$$

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-1\right)^{3}.

$$6x^{2}+12x+4+2x^{3}-2\left(x^{3}-3x^{2}+3x-1\right)=32$$

Use the distributive property to multiply -2 by x^{3}-3x^{2}+3x-1.

$$6x^{2}+12x+4+2x^{3}-2x^{3}+6x^{2}-6x+2=32$$

Combine 2x^{3} and -2x^{3} to get 0.

$$6x^{2}+12x+4+6x^{2}-6x+2=32$$

Combine 6x^{2} and 6x^{2} to get 12x^{2}.

$$12x^{2}+12x+4-6x+2=32$$

Combine 12x and -6x to get 6x.

$$12x^{2}+6x+4+2=32$$

Add 4 and 2 to get 6.

$$12x^{2}+6x+6=32$$

Subtract 32 from both sides.

$$12x^{2}+6x+6-32=0$$

Subtract 32 from 6 to get -26.

$$12x^{2}+6x-26=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, 6 for b, and -26 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\times 12\left(-26\right)}}{2\times 12}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\times 12\left(-26\right)}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-6±\sqrt{36-48\left(-26\right)}}{2\times 12}$$

Multiply -48 times -26.

$$x=\frac{-6±\sqrt{36+1248}}{2\times 12}$$

Add 36 to 1248.

$$x=\frac{-6±\sqrt{1284}}{2\times 12}$$

Take the square root of 1284.

$$x=\frac{-6±2\sqrt{321}}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{-6±2\sqrt{321}}{24}$$

Now solve the equation x=\frac{-6±2\sqrt{321}}{24} when ± is plus. Add -6 to 2\sqrt{321}.

$$x=\frac{2\sqrt{321}-6}{24}$$

Divide -6+2\sqrt{321} by 24.

$$x=\frac{\sqrt{321}}{12}-\frac{1}{4}$$

Now solve the equation x=\frac{-6±2\sqrt{321}}{24} when ± is minus. Subtract 2\sqrt{321} from -6.

$$x=\frac{-2\sqrt{321}-6}{24}$$

Divide -6-2\sqrt{321} by 24.

$$x=-\frac{\sqrt{321}}{12}-\frac{1}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{321}}{12}-\frac{1}{4} x=-\frac{\sqrt{321}}{12}-\frac{1}{4}$$

Steps for Completing the Square

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

$$6\left(x^{2}+2x+1\right)+2\left(x-1\right)\left(x^{2}+x+1\right)-2\left(x-1\right)^{3}=32$$

Use the distributive property to multiply 6 by x^{2}+2x+1.

$$6x^{2}+12x+6+2\left(x-1\right)\left(x^{2}+x+1\right)-2\left(x-1\right)^{3}=32$$

Use the distributive property to multiply 2 by x-1.

$$6x^{2}+12x+6+\left(2x-2\right)\left(x^{2}+x+1\right)-2\left(x-1\right)^{3}=32$$

Use the distributive property to multiply 2x-2 by x^{2}+x+1 and combine like terms.

$$6x^{2}+12x+6+2x^{3}-2-2\left(x-1\right)^{3}=32$$

Subtract 2 from 6 to get 4.

$$6x^{2}+12x+4+2x^{3}-2\left(x-1\right)^{3}=32$$

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-1\right)^{3}.

$$6x^{2}+12x+4+2x^{3}-2\left(x^{3}-3x^{2}+3x-1\right)=32$$

Use the distributive property to multiply -2 by x^{3}-3x^{2}+3x-1.

$$6x^{2}+12x+4+2x^{3}-2x^{3}+6x^{2}-6x+2=32$$

Combine 2x^{3} and -2x^{3} to get 0.

$$6x^{2}+12x+4+6x^{2}-6x+2=32$$

Combine 6x^{2} and 6x^{2} to get 12x^{2}.

$$12x^{2}+12x+4-6x+2=32$$

Combine 12x and -6x to get 6x.

$$12x^{2}+6x+4+2=32$$

Add 4 and 2 to get 6.

$$12x^{2}+6x+6=32$$

Subtract 6 from both sides.

$$12x^{2}+6x=32-6$$

Subtract 6 from 32 to get 26.

$$12x^{2}+6x=26$$

Divide both sides by 12.

$$\frac{12x^{2}+6x}{12}=\frac{26}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}+\frac{6}{12}x=\frac{26}{12}$$

Reduce the fraction \frac{6}{12} to lowest terms by extracting and canceling out 6.

$$x^{2}+\frac{1}{2}x=\frac{26}{12}$$

Reduce the fraction \frac{26}{12} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{1}{2}x=\frac{13}{6}$$

Divide \frac{1}{2}, the coefficient of the x term, by 2 to get \frac{1}{4}. Then add the square of \frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{2}x+\left(\frac{1}{4}\right)^{2}=\frac{13}{6}+\left(\frac{1}{4}\right)^{2}$$

Square \frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=\frac{13}{6}+\frac{1}{16}$$

Add \frac{13}{6} to \frac{1}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=\frac{107}{48}$$

Factor x^{2}+\frac{1}{2}x+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{4}\right)^{2}=\frac{107}{48}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{4}\right)^{2}}=\sqrt{\frac{107}{48}}$$

Simplify.

$$x+\frac{1}{4}=\frac{\sqrt{321}}{12} x+\frac{1}{4}=-\frac{\sqrt{321}}{12}$$

Subtract \frac{1}{4} from both sides of the equation.

$$x=\frac{\sqrt{321}}{12}-\frac{1}{4} x=-\frac{\sqrt{321}}{12}-\frac{1}{4}$$