Solve left(4+2xright)^2={left(2sqrt{x^2+1}right)}^2

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

Solve for x

$$x=-\frac{3}{4}=-0.75$$

Steps for Solving Linear Equation

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

$$16+16x+4x^{2}=\left(2\sqrt{x^{2}+1}\right)^{2}$$

Expand \left(2\sqrt{x^{2}+1}\right)^{2}.

$$16+16x+4x^{2}=2^{2}\left(\sqrt{x^{2}+1}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$16+16x+4x^{2}=4\left(\sqrt{x^{2}+1}\right)^{2}$$

Calculate \sqrt{x^{2}+1} to the power of 2 and get x^{2}+1.

$$16+16x+4x^{2}=4\left(x^{2}+1\right)$$

Use the distributive property to multiply 4 by x^{2}+1.

$$16+16x+4x^{2}=4x^{2}+4$$

Subtract 4x^{2} from both sides.

$$16+16x+4x^{2}-4x^{2}=4$$

Combine 4x^{2} and -4x^{2} to get 0.

$$16+16x=4$$

Subtract 16 from both sides.

$$16x=4-16$$

Subtract 16 from 4 to get -12.

$$16x=-12$$

Divide both sides by 16.

$$x=\frac{-12}{16}$$

Reduce the fraction \frac{-12}{16} to lowest terms by extracting and canceling out 4.

$$x=-\frac{3}{4}$$