This question tests finding factors of a quadratic polynomial. To find factors of h(x) = x² + 2x - 15, we need to factor the expression. We look for two numbers that multiply to -15 and add to 2; these are 5 and -3. Therefore, h(x) = (x + 5)(x - 3), giving us factors (x + 5) and (x - 3). Choice C correctly identifies (x + 5) as a factor. A common mistake is to confuse the signs in factoring, potentially selecting (x - 5) or mixing up which values work.

This question tests quadratic polynomial relationships by asking us to solve a quadratic equation. To find possible values of x, we need to factor the quadratic expression x² - 5x + 6 = 0. We look for two numbers that multiply to 6 and add to -5; these are -2 and -3. Therefore, x² - 5x + 6 = (x - 2)(x - 3) = 0, giving us x = 2 or x = 3. The correct answer is B, which gives x = 2 as a possible value. A common error would be to confuse the signs when factoring, leading to incorrect values like x = -2.

This question tests polynomial factorization, specifically recognizing a difference of squares pattern. The expression x² - 9 follows the pattern a² - b² = (a + b)(a - b), where a = x and b = 3. Applying this formula, we get x² - 9 = x² - 3² = (x + 3)(x - 3). Therefore, the correct factorization is (x - 3)(x + 3), which matches choice C. A common mistake is to factor this as (x - 3)², which would expand to x² - 6x + 9, not x² - 9.

This question tests understanding of quadratic function properties, particularly the effect of the leading coefficient. In the function g(x) = -2x² + 8x + 3, the coefficient of x² is -2, which is negative. When the leading coefficient of a quadratic is negative, the parabola opens downward, making choice B correct. The y-intercept occurs when x = 0, giving g(0) = 3, not 8 as choice C suggests. Choice A incorrectly states that the parabola opens upward despite acknowledging the negative coefficient.

This question tests evaluating polynomial functions and computing differences. To find f(5) - f(-1), we must evaluate f(x) = x² - 4x - 5 at both x = 5 and x = -1. For f(5): f(5) = 5² - 4(5) - 5 = 25 - 20 - 5 = 0. For f(-1): f(-1) = (-1)² - 4(-1) - 5 = 1 + 4 - 5 = 0. Therefore, f(5) - f(-1) = 0 - 0 = 0, giving us answer A. A common error would be to make arithmetic mistakes when substituting negative values, potentially leading to incorrect answers like -20 or 20.

This question tests understanding of polynomial zeros from factored form. When a polynomial is written as p(x) = (x - 1)(x + 4)(x - 2), the zeros occur when any factor equals zero. Setting each factor to zero: x - 1 = 0 gives x = 1; x + 4 = 0 gives x = -4; x - 2 = 0 gives x = 2. Therefore, the polynomial has zeros at x = 1, -4, and 2, which matches choice D. Choice A incorrectly identifies the zero from (x + 4) as x = 4 instead of x = -4, a common sign error when working with factored forms.

This question tests finding the maximum of a quadratic function in a real-world context. For a quadratic function in the form H(t) = at² + bt + c with a < 0, the maximum occurs at t = -b/(2a). In H(t) = -5t² + 20t + 1, we have a = -5 and b = 20. Therefore, the maximum occurs at t = -20/(2(-5)) = -20/(-10) = 2 seconds. The answer is B. A common error would be to use the wrong formula or make sign errors in the calculation, potentially getting t = 4 or another incorrect value.

This question tests evaluating a cubic polynomial at a specific value. To find q(3) where q(x) = x³ - 3x² - 4x + 12, we substitute x = 3 into the expression. Computing step by step: q(3) = 3³ - 3(3²) - 4(3) + 12 = 27 - 3(9) - 12 + 12 = 27 - 27 - 12 + 12 = 0. Therefore, q(3) = 0, making A the correct answer. A common error would be arithmetic mistakes in computing powers or handling negative terms, potentially leading to answers like 6 or -6.

This question explores properties of quadratic graphs based on coefficients. For y = ax² + bx + c with a>0 and c<0, the parabola opens upward, and the y-intercept is c, which is negative. This means the graph crosses the y-axis below the x-axis. The sign of a determines the direction, and c directly gives the y-intercept. This justifies choice D as true. A distractor like choice A fails because a>0 means it opens upward, not downward, a sign misinterpretation. Another, like choice B, is not necessarily true since the discriminant could allow real intercepts.

This question examines properties of quadratic polynomials from coefficients. For q(x) = -2x² + 7x - 3, the leading coefficient -2 is negative, so the parabola opens downward. This determines the end behavior for large |x|. The y-intercept is -3, not 7, and axis is x = -7/(2*(-2)) = 7/4. This justifies choice B as true. A distractor like choice D fails because the y-intercept is the constant term -3, possibly confusing with the linear coefficient. Another, like choice A, errs on the sign of the leading coefficient.