1. The problem is to find the Laplace transform of the sum of two functions, say $f(t)$ and $g(t)$. 2. Recall the linearity property of the Laplace transform: $$\mathcal{L}\{f(t) + g(t)\} = \mathcal{L}\{f(t)\} + \mathcal{L}\{g(t)\}.$$ 3. Suppose $f(t) = t$ and $g(t) = e^{2t}$. We want to find $$\mathcal{L}\{t + e^{2t}\}.$$ 4. Find the Laplace transform of each function separately: - For $f(t) = t$, $$\mathcal{L}\{t\} = \frac{1}{s^2}.$$ - For $g(t) = e^{2t}$, $$\mathcal{L}\{e^{2t}\} = \frac{1}{s - 2}, \quad \text{for } s > 2.$$ 5. Using linearity, sum the transforms: $$\mathcal{L}\{t + e^{2t}\} = \frac{1}{s^2} + \frac{1}{s - 2}.$$ 6. This is the Laplace transform of the sum $t + e^{2t}$.