1. The problem is to factor the quadratic expressions and verify the factorizations given. 2. For the first expression $6x^2 + 9x + 3$, we look for common factors first. 3. Factor out the greatest common factor (GCF): $$6x^2 + 9x + 3 = 3(2x^2 + 3x + 1)$$ 4. Now factor the quadratic inside the parentheses: We look for two numbers that multiply to $2 \times 1 = 2$ and add to $3$. These numbers are $2$ and $1$. 5. Rewrite the middle term: $$3(2x^2 + 2x + x + 1)$$ 6. Group terms: $$3((2x^2 + 2x) + (x + 1))$$ 7. Factor each group: $$3(2x(x + 1) + 1(x + 1))$$ 8. Factor out the common binomial: $$3(x + 1)(2x + 1)$$ 9. So the factorization of $6x^2 + 9x + 3$ is: $$3(x + 1)(2x + 1)$$ 10. The given factorization $3(x + 2)(x + 1)$ is incorrect because $(x + 2)(x + 1) = x^2 + 3x + 2$, which does not match the quadratic inside the parentheses. 11. For the second expression $x^2 + 5x + 6$, factor it by finding two numbers that multiply to $6$ and add to $5$. These numbers are $2$ and $3$. 12. So the factorization is: $$ (x + 2)(x + 3) $$ 13. The given factorization $(x + 6)(x - 1)$ expands to $x^2 + 5x - 6$, which is incorrect. 14. Therefore, the correct factorizations are: - $6x^2 + 9x + 3 = 3(x + 1)(2x + 1)$ - $x^2 + 5x + 6 = (x + 2)(x + 3)$ 15. The factorizations you wrote are not correct.