This question tests ISEE Upper Level Mathematics Achievement skills, specifically factoring quadratic trinomials. Factoring involves breaking down an expression into a product of simpler expressions, which is essential for simplifying expressions, solving equations, and understanding polynomial functions. For x² - 11x + 30, we need to find two numbers that multiply to give +30 and add to give -11, which are -5 and -6 since (-5) × (-6) = 30 and (-5) + (-6) = -11. Choice A, (x-5)(x-6), is correct because when expanded using FOIL, we get x² - 6x - 5x + 30 = x² - 11x + 30, matching the original expression. Choice C is incorrect because while (-3) × (-10) = 30, the sum (-3) + (-10) = -13, not -11, demonstrating a common error where students find factor pairs of the constant term but don't verify that they sum to the correct middle coefficient. Teaching strategies include systematically listing all factor pairs of the constant term, checking both the product and sum requirements, and using area models or algebra tiles to visualize the factoring process.

This question tests ISEE Upper Level Mathematics Achievement skills, specifically identifying the first step in factoring expressions. Factoring involves breaking down an expression into a product of simpler expressions, and the first step should always be to look for a greatest common factor (GCF). For the expression 3x² + 12x, both terms share common factors: each term is divisible by 3 and by x, making the GCF equal to 3x. Choice A, "Factor out the GCF 3x," is correct because this is the standard first step in any factoring problem - we extract 3x to get 3x(x + 4). Choice B is incorrect because difference of squares doesn't apply to this expression (it's not in the form a² - b²), demonstrating a common error where students try to apply special factoring patterns before checking for common factors. Teaching strategies include creating a systematic factoring checklist with GCF as step one, practicing identification of common factors in each term, and emphasizing that factoring out the GCF often makes subsequent factoring steps easier and more obvious.

This question tests ISEE Upper Level Mathematics Achievement skills, specifically factoring difference of squares. Factoring involves breaking down an expression into a product of simpler expressions, which is essential for simplifying expressions, solving equations, and understanding polynomial functions. For the given expression x² - 9, we recognize this as a difference of squares pattern a² - b² = (a-b)(a+b), where a = x and b = 3. Choice B, (x-3)(x+3), is correct because it follows the difference of squares formula, and when expanded using FOIL gives x² + 3x - 3x - 9 = x² - 9. Choice A is incorrect because (x-3)(x-3) = x² - 6x + 9, representing a perfect square trinomial rather than a difference of squares, a common error where students confuse these two special factoring patterns. Teaching strategies include memorizing the difference of squares pattern, using area models to visualize why (a-b)(a+b) = a² - b², and practicing recognition of perfect squares like 1, 4, 9, 16, 25 to quickly identify when this pattern applies.

This question tests ISEE Upper Level Mathematics Achievement skills, specifically factoring difference of squares with coefficients. Factoring involves breaking down an expression into a product of simpler expressions, which is essential for simplifying expressions and understanding polynomial functions. For 4x² - 25, we recognize this as a difference of squares where 4x² = (2x)² and 25 = 5², following the pattern a² - b² = (a-b)(a+b). Choice A, (2x-5)(2x+5), is correct because it applies the difference of squares formula with a = 2x and b = 5, and when expanded gives 4x² + 10x - 10x - 25 = 4x² - 25. Choice C is incorrect because (2x-5)² = 4x² - 20x + 25, which is a perfect square trinomial, not a difference of squares, showing confusion between these two important factoring patterns. Teaching strategies include recognizing perfect squares of expressions like (2x)² = 4x², using the FOIL method to verify factorizations, and creating visual models that show why the middle terms cancel in difference of squares.

This question tests ISEE Upper Level Mathematics Achievement skills, specifically factoring quadratic equations to find roots. Factoring involves breaking down an expression into a product of simpler expressions, which is essential for solving equations and understanding polynomial functions. For the equation x² - 5x - 14 = 0, we need to find two numbers that multiply to -14 and add to -5, which are -7 and 2, giving us (x - 7)(x + 2) = 0. Choice A, x = 7, -2, is correct because when we set each factor equal to zero, we get x - 7 = 0 (so x = 7) and x + 2 = 0 (so x = -2). Choice B is incorrect because it reverses the signs of the roots, a common error where students confuse the relationship between the factors and the roots, forgetting that (x - a) = 0 means x = a, not x = -a. Teaching strategies include using the zero product property explicitly, checking answers by substitution back into the original equation, and creating a systematic approach for finding factor pairs of the constant term.

This question tests ISEE Upper Level Mathematics Achievement skills, specifically factoring quadratic expressions. Factoring involves breaking down an expression into a product of simpler expressions, which is essential for simplifying expressions, solving equations, and understanding polynomial functions. For the given expression x² + 7x + 12, the correct factors are derived by identifying pairs of numbers that multiply to give the constant term (12) and add to give the linear coefficient (7). Choice A, (x+3)(x+4), is correct because 3 × 4 = 12 and 3 + 4 = 7, and when expanded gives x² + 4x + 3x + 12 = x² + 7x + 12. Choice B is incorrect because it would expand to x² - 7x + 12, showing a sign error that students make when they don't carefully check whether factors should be positive or negative. Teaching strategies include using the FOIL method to verify answers, creating factor trees for the constant term, and emphasizing the relationship between the signs in the original expression and the factored form.

This question tests ISEE Upper Level Mathematics Achievement skills, specifically factoring expressions with common factors. Factoring involves breaking down an expression into a product of simpler expressions, which is essential for simplifying expressions, solving equations, and understanding polynomial functions. For the given expression 2x² - 10x, the correct approach is to identify the greatest common factor (GCF) of both terms, which is 2x. Choice A, 2x(x-5), is correct because when we factor out 2x from each term, we get 2x(x) - 2x(5) = 2x(x - 5), which equals the original expression. Choice B is incorrect because while it equals the original expression, it only factors out x instead of the full GCF of 2x, demonstrating incomplete factoring where students miss extracting all common factors. Teaching strategies include listing all factors of each term to identify the GCF, emphasizing that we should always factor out the greatest common factor first, and checking work by distributing to verify the factored form equals the original expression.