Answer
$$\tan\theta=\frac{2\sqrt{2(x+2)}}{|x|}$$
Work Step by Step
$$\sec\theta=\frac{x+4}{x}$$
According to the Pythagorean Identity
$$\tan^2\theta+1=\sec^2\theta$$
we can rewrite as $$\tan^2\theta=\sec^2\theta-1$$
Then, $$\tan\theta=\sqrt{\tan^2\theta}$$
This is the way we would be able to find an expression of $x$ for $\tan\theta$
1) Find $\sec^2\theta$
$$\sec^2\theta=\frac{(x+4)^2}{x^2}$$
2) Find $\tan^2\theta$
$$\tan^2\theta=\sec^2\theta-1$$
$$\tan^2\theta=\frac{(x+4)^2}{x^2}-1$$
$$\tan^2\theta=\frac{(x+4)^2-x^2}{x^2}$$
$$\tan^2\theta=\frac{(x+4-x)(x+4+x)}{x^2}$$ (for $a^2-b^2=(a-b)(a+b)$)
$$\tan^2\theta=\frac{4(2x+4)}{x^2}$$
$$\tan^2\theta=\frac{8(x+2)}{x^2}$$
3) Find $\tan\theta$
$$\tan\theta=\sqrt{\tan^2\theta}$$
$$\tan\theta=\frac{\sqrt{8(x+2)}}{\sqrt{x^2}}$$
$$\tan\theta=\frac{2\sqrt{2(x+2)}}{|x|}$$ (since $\sqrt{a^2}=|a|$)
Unfortunately, we cannot eliminate the absolute value $||$ sign, since there is not enough information to know whether $x\gt0$ or $\lt0$.
