1. **Problem statement:** Find the Laplace transform of the function $f(t) = 4t - 10$ using appropriate properties. 2. **Recall the Laplace transform formulas:** - $\mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}$ for $n$ a non-negative integer. - $\mathcal{L}\{c\} = \frac{c}{s}$ for constant $c$. - The Laplace transform is linear: $\mathcal{L}\{af(t) + bg(t)\} = a\mathcal{L}\{f(t)\} + b\mathcal{L}\{g(t)\}$. 3. **Apply linearity:** $$\mathcal{L}\{4t - 10\} = 4\mathcal{L}\{t\} - 10\mathcal{L}\{1\}$$ 4. **Use formulas:** - $\mathcal{L}\{t\} = \frac{1}{s^2}$ since $n=1$ and $1! = 1$. - $\mathcal{L}\{1\} = \frac{1}{s}$. 5. **Substitute:** $$4 \times \frac{1}{s^2} - 10 \times \frac{1}{s} = \frac{4}{s^2} - \frac{10}{s}$$ 6. **Final answer:** $$\boxed{\mathcal{L}\{4t - 10\} = \frac{4}{s^2} - \frac{10}{s}}$$ This completes the solution for part (a).