Solve a(x^2+x+1)-(a+10)x-12=0

$$a ( x ^ { 2 } + x + 1 ) - ( a + 10 ) x - 12 = 0$$

Solve for a (complex solution)

$$a=\frac{2\left(5x+6\right)}{x^{2}+1}$$

$$x\neq -i\text{ and }x\neq i$$

Steps for Solving Linear Equation

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

$$ax^{2}+ax+a-\left(a+10\right)x-12=0$$

Use the distributive property to multiply a+10 by x.

$$ax^{2}+ax+a-\left(ax+10x\right)-12=0$$

To find the opposite of ax+10x, find the opposite of each term.

$$ax^{2}+ax+a-ax-10x-12=0$$

Combine ax and -ax to get 0.

$$ax^{2}+a-10x-12=0$$

Add 10x to both sides. Anything plus zero gives itself.

$$ax^{2}+a-12=10x$$

Add 12 to both sides.

$$ax^{2}+a=10x+12$$

Combine all terms containing a.

$$\left(x^{2}+1\right)a=10x+12$$

Divide both sides by x^{2}+1.

$$\frac{\left(x^{2}+1\right)a}{x^{2}+1}=\frac{10x+12}{x^{2}+1}$$

Dividing by x^{2}+1 undoes the multiplication by x^{2}+1.

$$a=\frac{10x+12}{x^{2}+1}$$

Divide 10x+12 by x^{2}+1.

$$a=\frac{2\left(5x+6\right)}{x^{2}+1}$$

Solve for a

$$a=\frac{2\left(5x+6\right)}{x^{2}+1}$$

Steps for Solving Linear Equation

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

$$ax^{2}+ax+a-\left(a+10\right)x-12=0$$

Use the distributive property to multiply a+10 by x.

$$ax^{2}+ax+a-\left(ax+10x\right)-12=0$$

To find the opposite of ax+10x, find the opposite of each term.

$$ax^{2}+ax+a-ax-10x-12=0$$

Combine ax and -ax to get 0.

$$ax^{2}+a-10x-12=0$$

Add 10x to both sides. Anything plus zero gives itself.

$$ax^{2}+a-12=10x$$

Add 12 to both sides.

$$ax^{2}+a=10x+12$$

Combine all terms containing a.

$$\left(x^{2}+1\right)a=10x+12$$

Divide both sides by x^{2}+1.

$$\frac{\left(x^{2}+1\right)a}{x^{2}+1}=\frac{10x+12}{x^{2}+1}$$

Dividing by x^{2}+1 undoes the multiplication by x^{2}+1.

$$a=\frac{10x+12}{x^{2}+1}$$

Divide 10x+12 by x^{2}+1.

$$a=\frac{2\left(5x+6\right)}{x^{2}+1}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{25+12a-a^{2}}+5}{a}\text{; }x=\frac{-\sqrt{25+12a-a^{2}}+5}{a}\text{, }&a\neq 0\\x=-\frac{6}{5}\text{, }&a=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{25+12a-a^{2}}+5}{a}\text{; }x=\frac{-\sqrt{25+12a-a^{2}}+5}{a}\text{, }&a\neq 0\text{ and }a\geq 6-\sqrt{61}\text{ and }a\leq \sqrt{61}+6\\x=-\frac{6}{5}\text{, }&a=0\end{matrix}\right.$$