1. Problem: Check for mistakes in the given algebraic expansions and factorizations. 2. We will verify each item by expanding or factoring and comparing with the provided expressions. 3. Item (19) (2x + 3)(3x + 2): Expand: $$ (2x)(3x) + (2x)(2) + (3)(3x) + (3)(2) = 6x^2 + 4x + 9x + 6 = 6x^2 + 13x + 6 $$ Provided: $$ 6x^2 + 21x + 9x + 6 $$ and $$ 6x^2 + 18x + 6 $$ Mistake: Yes, the middle terms are incorrect; correct middle term is $13x$. 4. Item (24) (a^3 + 4)(a^3 - 9): Expand: $$ a^6 - 9a^3 + 4a^3 - 36 = a^6 - 5a^3 - 36 $$ Provided matches correct expansion. Mistake: No. 5. Item (25) (6x - 7y)(5x + 4y): Expand: $$ 30x^2 + 24xy - 35xy - 28y^2 = 30x^2 - 11xy - 28y^2 $$ Provided: $$ 30x^2 - 9xy - 28y^2 $$ Mistake: Yes, middle term coefficient is wrong; should be $-11xy$. 6. Item (23) (a^3 + 3)(a^3 - 5): Expand: $$ a^6 - 5a^3 + 3a^3 - 15 = a^6 - 2a^3 - 15 $$ Provided matches correct expansion. Mistake: No. 7. Item (18) (x + 3)(x^2 - 4x + 3): Expand: $$ x^3 - 4x^2 + 3x + 3x^2 - 12x + 9 = x^3 - x^2 - 9x + 9 $$ Provided: $$ x^3 - x^2 - 12 $$ Mistake: Yes, missing $-9x + 9$ terms. 8. Item (16) (x - 3)(x + 4): Expand: $$ x^2 + 4x - 3x - 12 = x^2 + x - 12 $$ Provided: $$ x^2 + 4x - 3x - 12 $$ and simplified to $$ x^2 + x - 12 $$ Mistake: No. 9. Item (22) (7 - x)(6 - 3x): Expand: $$ 42 - 21x - 6x + 3x^2 = 42 - 27x + 3x^2 $$ Provided: $$ 42 - 21x - 6x - 4x^2 $$ and $$ 4x^2 - 27x + 42 $$ Mistake: Yes, coefficient of $x^2$ is incorrect; should be $+3x^2$. 10. Item (20) (5x - 4)(2x + 1): Expand: $$ 10x^2 + 5x - 8x - 4 = 10x^2 - 3x - 4 $$ Provided: $$ 10x^2 + 5x - 8x - 4 $$ Mistake: No, but final simplification missing. 11. Item (17) (5 - 3x)(4 + 2x): Expand: $$ 20 + 10x - 12x - 6x^2 = 20 - 2x - 6x^2 $$ Provided: $$ 6x^2 - 2x + 20 $$ Mistake: Yes, sign of $x^2$ term is wrong; should be $-6x^2$. 12. Item (21) (x^3)(x - 4): Expand: $$ x^4 - 4x^3 $$ Provided: $$ 10x^2 - 3x - 4 $$ Mistake: Yes, provided expression unrelated to expansion. 13. Item (19) (x - 3)(x + 4): Same as item (16), no mistake. Summary of items with mistakes: (19) first occurrence, (25), (18), (22), (17), (21)