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: Calculate the expansion of (x - 2)(x - 3).

Answer: x^2 - 5x + 6

Question 4: Expand: (3x + 4)(x - 5).

Answer: 3x^2 - 11x - 20

Question 1: If (x - 2)(x + 5) = 0, find the possible values of x. Explain your answer.

Answer: x = 2 or x = -5

Question 2: Expand (x - 7)(x + 3) and interpret the sign of the middle term.

Answer: x^2 - 4x - 21; middle term is negative because -7 + 3 = -4

Question 3: A rectangle's length and width are (x - 1) and (x + 2) respectively. Write an expression for its area and discuss how negatives might affect the size if x is negative.

Answer: Area = (x - 1)(x + 2); if x is negative, the area depends on the value of x but negative dimensions are not physically possible, so x must be ≥ 1 for positive area.

Question 4: Solve for x: (x - 4)(x + 6) = 0, and explain what the negative root signifies in context.

Answer: x = 4 or x = -6; negative root may represent a negative value not feasible in some contexts.

Question 1: A car's speed is modeled by (x - 3)(x + 2). If x = -1, what is the speed? Explain the significance of the negative x value.

Answer: Speed = (-1 - 3)(-1 + 2) = (-4)(1) = -4; negative speed indicates a direction opposite to the defined positive direction.

Question 2: A gardener plants (x - 2)(x + 4) trees along a straight line. If x is negative, what does that imply about the planting process?

Answer: If x is negative, it may suggest a conceptual or hypothetical scenario, as negative quantities are not physically meaningful in planting context.

Question 1: Expand and simplify: (x + 5)(x - 2) + (x - 3)(x + 4). Then analyze how negatives impact the overall expression.

Answer: x^2 + 3x - 10 + x^2 + x - 12 = 2x^2 + 4x - 22

Question 2: Find the value of x for which (x - 1)(x + 6) = (x + 2)(x - 4). Explain the role of negatives in your solution.

Answer: x^2 + 5x - 6 = x^2 - 2x - 8; 7x = -2; x = -2/7; negatives affect the sign of the linear terms, changing the solution.

Question 1: Identify the mistake in expanding: (x - 3)(x + 5) = x^2 + 8x + 15.

Answer: Incorrect; the middle term should be -3x + 5x = 2x, so the correct expansion is x^2 + 2x - 15.

Question 2: Write an expression for the product of (x - 2) and (x - 5), then expand it.

Answer: Expression: (x - 2)(x - 5); expansion: x^2 - 7x + 10

Question 1: A student expanded (x - 4)(x + 2) and wrote x^2 + 2x - 8, claiming the middle term is positive because both brackets are positive. Correct and explain the mistake.

Answer: Incorrect; (x - 4)(x + 2) expands to x^2 + 2x - 4x - 8 = x^2 - 2x - 8. The middle term is negative because -4 + 2 = -2.