1. **State the problem:** Factorize the expression $2y(x - 1) + 5y(x - 1)^3$ by removing common factors. 2. **Identify common factors:** Both terms contain the factor $y$ and $(x - 1)$, but the second term has $(x - 1)^3$ which can be written as $(x - 1) \cdot (x - 1)^2$. 3. **Extract the common factors:** The smallest power of $(x - 1)$ common to both terms is $(x - 1)^1$, so the common factor is $y(x - 1)$. 4. **Rewrite the expression:** $$ 2y(x - 1) + 5y(x - 1)^3 = y(x - 1) \left(2 + 5(x - 1)^2\right) $$ 5. **Final factored form:** $$ \boxed{y(x - 1) \left(2 + 5(x - 1)^2\right)} $$ This is the fully factored expression by removing the common factors $y(x - 1)$.