1. The problem asks to explain why the student's subtraction of $(3x^2 + 5y + 2) - (4x^2 + 3y + 2)$ is incorrect and to find the correct answer. 2. The student incorrectly treated subtraction of polynomials and misapplied the minus sign and exponents. The subtraction of $(4x^2 + 3y + 2)$ means subtracting each term inside the parentheses, not squaring or changing signs arbitrarily. 3. Correct subtraction formula: $ (A) - (B) = A - B = A + (-B) $. Distribute the minus sign to each term in $B$. 4. Perform the subtraction step-by-step: $$ (3x^2 + 5y + 2) - (4x^2 + 3y + 2) = 3x^2 + 5y + 2 - 4x^2 - 3y - 2 $$ 5. Combine like terms: $$ (3x^2 - 4x^2) + (5y - 3y) + (2 - 2) = -x^2 + 2y + 0 = -x^2 + 2y $$ 6. So, the correct answer is $ -x^2 + 2y $. --- 7. The difference between two polynomials is $5x + 3$. One polynomial is $4x + 1 - 3x^2$. Find the other polynomial. 8. Let the other polynomial be $P$. Then: $$ (4x + 1 - 3x^2) - P = 5x + 3 $$ 9. Rearranged: $$ P = (4x + 1 - 3x^2) - (5x + 3) = 4x + 1 - 3x^2 - 5x - 3 $$ 10. Combine like terms: $$ (4x - 5x) + (1 - 3) - 3x^2 = -x - 2 - 3x^2 = -3x^2 - x - 2 $$ 11. So, the other polynomial is $ -3x^2 - x - 2 $. --- 12. Subtract the following: a) $(mn - 5m - 7) - (-6n + 2m + 1)$ $$ = mn - 5m - 7 + 6n - 2m - 1 $$ $$ = mn - 7m + 6n - 8 $$ b) $(2a + 3b - 3a^2 + b^2) - (-a^2 + 8b^2 + 3a - b)$ $$ = 2a + 3b - 3a^2 + b^2 + a^2 - 8b^2 - 3a + b $$ $$ = (2a - 3a) + (3b + b) + (-3a^2 + a^2) + (b^2 - 8b^2) $$ $$ = -a + 4b - 2a^2 - 7b^2 $$ c) $(xy - x - 5y + 4y^2) - (6x^2 + 9y - xy)$ $$ = xy - x - 5y + 4y^2 - 6x^2 - 9y + xy $$ $$ = (xy + xy) - x - (5y + 9y) + 4y^2 - 6x^2 $$ $$ = 2xy - x - 14y + 4y^2 - 6x^2 $$