Solve (x+2)(x+3)(x+1)=(x+4)(x+4)(x-4)+7

$$( x + 2 ) ( x + 3 ) ( x + 1 ) = ( x + 4 ) ( x + 4 ) ( x - 4 ) + 7$$

Solve for x

$$x = -\frac{21}{2} = -10\frac{1}{2} = -10.5$$

$$x=-3$$

Steps Using the Quadratic Formula

Multiply x+4 and x+4 to get \left(x+4\right)^{2}.

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

Use the distributive property to multiply x+2 by x+3 and combine like terms.

$$\left(x^{2}+5x+6\right)\left(x+1\right)=\left(x+4\right)^{2}\left(x-4\right)+7$$

Use the distributive property to multiply x^{2}+5x+6 by x+1 and combine like terms.

$$x^{3}+6x^{2}+11x+6=\left(x+4\right)^{2}\left(x-4\right)+7$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+4\right)^{2}.

$$x^{3}+6x^{2}+11x+6=\left(x^{2}+8x+16\right)\left(x-4\right)+7$$

Use the distributive property to multiply x^{2}+8x+16 by x-4 and combine like terms.

$$x^{3}+6x^{2}+11x+6=x^{3}+4x^{2}-16x-64+7$$

Add -64 and 7 to get -57.

$$x^{3}+6x^{2}+11x+6=x^{3}+4x^{2}-16x-57$$

Subtract x^{3} from both sides.

$$x^{3}+6x^{2}+11x+6-x^{3}=4x^{2}-16x-57$$

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

$$6x^{2}+11x+6=4x^{2}-16x-57$$

Subtract 4x^{2} from both sides.

$$6x^{2}+11x+6-4x^{2}=-16x-57$$

Combine 6x^{2} and -4x^{2} to get 2x^{2}.

$$2x^{2}+11x+6=-16x-57$$

Add 16x to both sides.

$$2x^{2}+11x+6+16x=-57$$

Combine 11x and 16x to get 27x.

$$2x^{2}+27x+6=-57$$

Add 57 to both sides.

$$2x^{2}+27x+6+57=0$$

Add 6 and 57 to get 63.

$$2x^{2}+27x+63=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 27 for b, and 63 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-27±\sqrt{27^{2}-4\times 2\times 63}}{2\times 2}$$

Square 27.

$$x=\frac{-27±\sqrt{729-4\times 2\times 63}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-27±\sqrt{729-8\times 63}}{2\times 2}$$

Multiply -8 times 63.

$$x=\frac{-27±\sqrt{729-504}}{2\times 2}$$

Add 729 to -504.

$$x=\frac{-27±\sqrt{225}}{2\times 2}$$

Take the square root of 225.

$$x=\frac{-27±15}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-27±15}{4}$$

Now solve the equation x=\frac{-27±15}{4} when ± is plus. Add -27 to 15.

$$x=-\frac{12}{4}$$

Divide -12 by 4.

$$x=-3$$

Now solve the equation x=\frac{-27±15}{4} when ± is minus. Subtract 15 from -27.

$$x=-\frac{42}{4}$$

Reduce the fraction \frac{-42}{4} to lowest terms by extracting and canceling out 2.

$$x=-\frac{21}{2}$$

The equation is now solved.

$$x=-3 x=-\frac{21}{2}$$

Steps for Completing the Square

Multiply x+4 and x+4 to get \left(x+4\right)^{2}.

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

Use the distributive property to multiply x+2 by x+3 and combine like terms.

$$\left(x^{2}+5x+6\right)\left(x+1\right)=\left(x+4\right)^{2}\left(x-4\right)+7$$

Use the distributive property to multiply x^{2}+5x+6 by x+1 and combine like terms.

$$x^{3}+6x^{2}+11x+6=\left(x+4\right)^{2}\left(x-4\right)+7$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+4\right)^{2}.

$$x^{3}+6x^{2}+11x+6=\left(x^{2}+8x+16\right)\left(x-4\right)+7$$

Use the distributive property to multiply x^{2}+8x+16 by x-4 and combine like terms.

$$x^{3}+6x^{2}+11x+6=x^{3}+4x^{2}-16x-64+7$$

Add -64 and 7 to get -57.

$$x^{3}+6x^{2}+11x+6=x^{3}+4x^{2}-16x-57$$

Subtract x^{3} from both sides.

$$x^{3}+6x^{2}+11x+6-x^{3}=4x^{2}-16x-57$$

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

$$6x^{2}+11x+6=4x^{2}-16x-57$$

Subtract 4x^{2} from both sides.

$$6x^{2}+11x+6-4x^{2}=-16x-57$$

Combine 6x^{2} and -4x^{2} to get 2x^{2}.

$$2x^{2}+11x+6=-16x-57$$

Add 16x to both sides.

$$2x^{2}+11x+6+16x=-57$$

Combine 11x and 16x to get 27x.

$$2x^{2}+27x+6=-57$$

Subtract 6 from both sides.

$$2x^{2}+27x=-57-6$$

Subtract 6 from -57 to get -63.

$$2x^{2}+27x=-63$$

Divide both sides by 2.

$$\frac{2x^{2}+27x}{2}=-\frac{63}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{27}{2}x=-\frac{63}{2}$$

Divide \frac{27}{2}, the coefficient of the x term, by 2 to get \frac{27}{4}. Then add the square of \frac{27}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{27}{2}x+\left(\frac{27}{4}\right)^{2}=-\frac{63}{2}+\left(\frac{27}{4}\right)^{2}$$

Square \frac{27}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{27}{2}x+\frac{729}{16}=-\frac{63}{2}+\frac{729}{16}$$

Add -\frac{63}{2} to \frac{729}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{27}{2}x+\frac{729}{16}=\frac{225}{16}$$

Factor x^{2}+\frac{27}{2}x+\frac{729}{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}.

$$\left(x+\frac{27}{4}\right)^{2}=\frac{225}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{27}{4}\right)^{2}}=\sqrt{\frac{225}{16}}$$

Simplify.

$$x+\frac{27}{4}=\frac{15}{4} x+\frac{27}{4}=-\frac{15}{4}$$

Subtract \frac{27}{4} from both sides of the equation.

$$x=-3 x=-\frac{21}{2}$$