To solve the equation \( \tan(x + \pi) + 2\sin(x + \pi) = 0 \) on the interval \( [0, 2\pi) \), we will first utilize the periodic properties of the tangent and sine functions.
We know that:
- \( \tan(x + \pi) = \tan x \)
- \( \sin(x + \pi) = -\sin x \)
Substituting these into our equation, we get:
\[ \tan x - 2\sin x = 0 \]
This can be rewritten as:
\[ \tan x = 2\sin x \]
Using the definition of tangent, we substitute \( \tan x = \frac{\sin x}{\cos x} \) into the equation:
\[ \frac{\sin x}{\cos x} = 2\sin x \]
Assuming \(\sin x \neq 0\), we can divide both sides by \(\sin x\):
\[ \frac{1}{\cos x} = 2 \]
This simplifies to:
\[ \cos x = \frac{1}{2} \]
The cosine function equals \( \frac{1}{2} \) at:
\[ x = \frac{\pi}{3}, \quad x = \frac{5\pi}{3} \]
Next, we need to check the case when \( \sin x = 0 \):
\[ \sin x = 0 \Rightarrow x = 0, \pi \]
Thus, we have the candidate solutions:
\[ x = 0, \quad x = \frac{\pi}{3}, \quad x = \pi, \quad x = \frac{5\pi}{3} \]
Finally, we will confirm that all of our candidate solutions satisfy the original equation:
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For \( x = 0 \): \[ \tan(0 + \pi) + 2\sin(0 + \pi) = \tan \pi + 2\sin \pi = 0 + 0 = 0 \]
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For \( x = \frac{\pi}{3} \): \[ \tan\left(\frac{\pi}{3} + \pi\right) + 2\sin\left(\frac{\pi}{3} + \pi\right) = \tan\left(\frac{4\pi}{3}\right) + 2\sin\left(\frac{4\pi}{3}\right) = \tan\left(\frac{4\pi}{3}\right) + 2\left(-\frac{\sqrt{3}}{2}\right) \] \[ = \frac{1}{\sqrt{3}} - \sqrt{3} = \frac{1 - 3}{\sqrt{3}} = \frac{-2}{\sqrt{3}} \neq 0 \]
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For \( x = \pi \): \[ \tan(\pi + \pi) + 2\sin(\pi + \pi) = \tan(2\pi) + 2\sin(2\pi) = 0 + 0 = 0 \]
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For \( x = \frac{5\pi}{3} \): \[ \tan\left(\frac{5\pi}{3} + \pi\right) + 2\sin\left(\frac{5\pi}{3} + \pi\right) = \tan\left(\frac{8\pi}{3}\right) + 2\sin\left(\frac{8\pi}{3}\right) = \tan\left(\frac{2\pi}{3}\right) + 2\left(-\frac{\sqrt{3}}{2}\right) \] \[ = -\frac{1}{\sqrt{3}} - \sqrt{3} = \frac{-1 - 3}{\sqrt{3}} = \frac{-4}{\sqrt{3}} \neq 0 \]
Thus, the valid solutions to the equation \( \tan(x + \pi) + 2\sin(x + \pi) = 0 \) within the interval \( [0, 2\pi) \) are:
\[ \boxed{0, \pi} \]