1. The problem is to simplify the expression $\cos^3 x + \cos^4 x$. 2. Factor out the common term $\cos^3 x$ from both parts: $$\cos^3 x + \cos^4 x = \cos^3 x (1 + \cos x)$$ 3. This is the simplified factorized form of the given expression. Therefore, the expression $\cos^3 x + \cos^4 x$ is equal to: $$\cos^3 x (1 + \cos x)$$