Given the polynomial function P(x) = (2x - 6)(x + 4) - (x - 3)2 + (5x - 15).

Here's the solution for question 2. Given the polynomial function P(x) = (2x - 6)(x + 4) - (x - 3)^2 + (5x - 15). a) Write P(x) as a product of its linear factors. Step 1: Factor out common terms from each part of the expression. Notice that (2x - 6) = 2(x - 3) and (5x - 15) = 5(x - 3). Substitute these into the expression for P(x): P(x) = 2(x - 3)(x + 4) - (x - 3)^2 + 5(x - 3) Step 2: Factor out the common linear factor (x - 3) from all terms. P(x) = (x - 3) [2(x + 4) - (x - 3) + 5] Step 3: Simplify the expression inside the square brackets. P(x) = (x - 3) [2x + 8 - x + 3 + 5] P(x) = (x - 3) [x + 16] The polynomial P(x) as a product of its linear factors is (x - 3)(x + 16). b) Expand, simplify and arrange P(x) in descending powers of x. Step 1: Use the factored form from part a) and expand it. P(x) = (x - 3)(x + 16) P(x) = x(x + 16) - 3(x + 16) P(x) = x^2 + 16x - 3x - 48 Step 2: Combine like terms. P(x) = x^2 + 13x - 48 The polynomial P(x) in descending powers of x is x^2 + 13x - 48. c) State the degree of P(x). The degree of a polynomial is the highest power of the variable in the polynomial. From part b), P(x) = x^2 + 13x - 48. The highest power of x is 2. The degree of P(x) is 2. d) State the domain of definition of P(x). P(x) is a polynomial function. Polynomial functions are defined for all real numbers. The domain of P(x) is R or (-, ). e) Solve in R the equation (x - 3)(x + 16) = 0. Step 1: Set each linear factor equal to zero. x - 3 = 0 or x + 16 = 0 Step 2: Solve for x in each equation. x = 3 or x = -16 The solutions are x = 3, x = -16. f) Determine the range of values of x for which (x - 3)(x + 16) 0. Step 1: Identify the roots of the quadratic expression. The roots are x = 3 and x = -16, as found in part e). Step 2: Analyze the sign of the quadratic expression. The expression (x - 3)(x + 16) is a parabola opening upwards (since the coefficient of x^2 is positive when expanded). This means the parabola is above or on the x-axis when x is less than or equal to the smaller root, or greater than or equal to the larger root. So, we need x -16 or x 3. The range of values for x is x -16 or x 3. In interval notation, this is (-, -16] [3, ). 3 done, 2 left today. You're making progress.