Question 1: Expand the expression: (x - 3)(x + 4).

Answer: x^2 + x - 12

Question 2: Expand and simplify: (2x - 5)(x - 2).

Answer: 2x^2 - 9x + 10

Question 3: Expand: (-x + 7)(x + 3).

Answer: -x^2 + 4x + 21

Question 4: Simplify: (3x - 2)(-x + 5).

Answer: -3x^2 + 17x - 10

Question 1: Expand and simplify: (x - 2)(x + 5). Then explain why ignoring the negative sign in the first term leads to an incorrect answer.

Answer: Correct expansion: x^2 + 3x - 10. Ignoring the negative sign would give x^2 + 7x, which misses the constant term and the correct middle term.

Question 2: A number is multiplied by (x - 4). When expanding (x - 4)(x + 6), what common mistake might students make involving negatives? How can it be avoided?

Answer: A common mistake is to forget to distribute the negative sign properly, leading to x^2 + 6x - 4x - 24 = x^2 + 2x - 24. To avoid this, students should carefully distribute and keep track of negatives when expanding.

Question 1: A rectangular garden has length (x - 3) meters and width (x + 2) meters. Write an expression for its area, expand it, and interpret the negative coefficient if any.

Answer: Area = (x - 3)(x + 2) = x^2 + 2x - 3x - 6 = x^2 - x - 6. The negative coefficient of x indicates the linear change in area as x increases.

Question 1: Expand: (-2x + 5)(3x - 4). Then, verify the expansion by multiplying out the terms carefully.

Answer: -6x^2 + 8x + 15x - 20 = -6x^2 + 23x - 20

Question 2: Given the expression (x - 7)(-x + 3), expand and determine whether the resulting quadratic has a positive or negative leading coefficient and explain why.

Answer: -x^2 + 4x - 21. The leading coefficient is negative because the x^2 term is multiplied by -1.

Question 1: Student's mistake: (x - 4)(x + 5) = x^2 + 9x. What is wrong? Correct the expansion.

Answer: The mistake is ignoring the negative sign and the distributive process. Correct expansion: x^2 + 5x - 4x - 20 = x^2 + x - 20.

Question 2: A student expands (x + 3)(x - 6) and writes the answer as x^2 - 18. Identify the mistake and give the correct expansion.

Answer: They forgot to include the middle term. Correct expansion: x^2 - 3x + 6x - 18 = x^2 + 3x - 18.