Solve 6+(a^2+1)x=(a^2+1)^2-2(a^2-2)

$$6 + ( a ^ { 2 } + 1 ) x = ( a ^ { 2 } + 1 ) ^ { 2 } - 2 ( a ^ { 2 } - 2 )$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=a^{2}-1\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&a=i\text{ or }a=-i\end{matrix}\right.$$

Steps for Solving Linear Equation

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

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

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a^{2}+1\right)^{2}.

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

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

Use the distributive property to multiply -2 by a^{2}-2.

$$6+a^{2}x+x=a^{4}+2a^{2}+1-2a^{2}+4$$

Combine 2a^{2} and -2a^{2} to get 0.

$$6+a^{2}x+x=a^{4}+1+4$$

Add 1 and 4 to get 5.

$$6+a^{2}x+x=a^{4}+5$$

Subtract 6 from both sides.

$$a^{2}x+x=a^{4}+5-6$$

Subtract 6 from 5 to get -1.

$$a^{2}x+x=a^{4}-1$$

Combine all terms containing x.

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

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

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

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

$$x=\frac{a^{4}-1}{a^{2}+1}$$

Divide a^{4}-1 by a^{2}+1.

$$x=a^{2}-1$$

Solve for x

$$x=a^{2}-1$$

Steps for Solving Linear Equation

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

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

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a^{2}+1\right)^{2}.

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

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

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

Use the distributive property to multiply -2 by a^{2}-2.

$$6+a^{2}x+x=a^{4}+2a^{2}+1-2a^{2}+4$$

Combine 2a^{2} and -2a^{2} to get 0.

$$6+a^{2}x+x=a^{4}+1+4$$

Add 1 and 4 to get 5.

$$6+a^{2}x+x=a^{4}+5$$

Subtract 6 from both sides.

$$a^{2}x+x=a^{4}+5-6$$

Subtract 6 from 5 to get -1.

$$a^{2}x+x=a^{4}-1$$

Combine all terms containing x.

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

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

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

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

$$x=\frac{a^{4}-1}{a^{2}+1}$$

Divide a^{4}-1 by a^{2}+1.

$$x=a^{2}-1$$

Solve for a (complex solution)

$$a=\sqrt{x+1}$$

$$a=-\sqrt{x+1}$$

$$a=-i$$

$$a=i$$

Solve for a

$$a=-\sqrt{x+1}$$

$$a=\sqrt{x+1}\text{, }x\geq -1$$