1. **State the problem:** Factorize the expression $2(x+a)^2 + 5(x+a) + 2$. 2. **Identify the structure:** Let $y = x + a$. The expression becomes $2y^2 + 5y + 2$. 3. **Use the quadratic factorization formula:** For $ay^2 + by + c$, find two numbers that multiply to $a \times c = 2 \times 2 = 4$ and add to $b = 5$. 4. **Find the numbers:** The numbers are 4 and 1 because $4 \times 1 = 4$ and $4 + 1 = 5$. 5. **Rewrite the middle term:** $2y^2 + 4y + y + 2$. 6. **Group terms:** $(2y^2 + 4y) + (y + 2)$. 7. **Factor each group:** $2y(y + 2) + 1(y + 2)$. 8. **Factor out the common binomial:** $(y + 2)(2y + 1)$. 9. **Substitute back $y = x + a$:** $(x + a + 2)(2(x + a) + 1)$. 10. **Simplify:** $(x + a + 2)(2x + 2a + 1)$. **Final answer:** $$ (x + a + 2)(2x + 2a + 1) $$