1. The problem is to simplify the expression $\tan\left(\frac{\pi}{4} + x\right)$.\n\n2. We use the tangent addition formula: $$\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}.$$\n\n3. Substitute $a = \frac{\pi}{4}$ and $b = x$: $$\tan\left(\frac{\pi}{4} + x\right) = \frac{\tan\frac{\pi}{4} + \tan x}{1 - \tan\frac{\pi}{4} \tan x}.$$\n\n4. Recall that $\tan\frac{\pi}{4} = 1$, so the expression becomes: $$\frac{1 + \tan x}{1 - 1 \cdot \tan x} = \frac{1 + \tan x}{1 - \tan x}.$$\n\n5. This is the simplified form of the expression.\n\nFinal answer: $$\tan\left(\frac{\pi}{4} + x\right) = \frac{1 + \tan x}{1 - \tan x}.$$