Solve left(x^2+1right)^2+6left(x^2+1right)+8=0

$${ \left( { x }^{ 2 } +1 \right) }^{ 2 } +6 \left( { x }^{ 2 } +1 \right) +8=0$$

Solve for x (complex solution)

$$x=\sqrt{3}i\approx 1.732050808i$$

$$x=-\sqrt{3}i\approx -0-1.732050808i$$

$$x=-\sqrt{5}i\approx -0-2.236067977i$$

$$x=\sqrt{5}i\approx 2.236067977i$$

Steps Using the Quadratic Formula

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

$$\left(x^{2}\right)^{2}+2x^{2}+1+6\left(x^{2}+1\right)+8=0$$

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

$$x^{4}+2x^{2}+1+6\left(x^{2}+1\right)+8=0$$

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

$$x^{4}+2x^{2}+1+6x^{2}+6+8=0$$

Combine 2x^{2} and 6x^{2} to get 8x^{2}.

$$x^{4}+8x^{2}+1+6+8=0$$

Add 1 and 6 to get 7.

$$x^{4}+8x^{2}+7+8=0$$

Add 7 and 8 to get 15.

$$x^{4}+8x^{2}+15=0$$

Substitute t for x^{2}.

$$t^{2}+8t+15=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, 8 for b, and 15 for c in the quadratic formula.

$$t=\frac{-8±\sqrt{8^{2}-4\times 1\times 15}}{2}$$

Do the calculations.

$$t=\frac{-8±2}{2}$$

Solve the equation t=\frac{-8±2}{2} when ± is plus and when ± is minus.

$$t=-3 t=-5$$

Since x=t^{2}, the solutions are obtained by evaluating x=±\sqrt{t} for each t.

$$x=-\sqrt{3}i x=\sqrt{3}i x=-\sqrt{5}i x=\sqrt{5}i$$