Let f(x) = ax3 - x2 + bx + 6. If (x+2) is a factor of f(x) and the remainder when f(x) is divided by (x+1) is 10, find the values of a and b. Then, find the roots of f(x)=0.

Here are the solutions to the problems you provided. Question 1: The text for question 1 is heavily garbled and unreadable. Therefore, it is not possible to provide a solution. Question 2: The polynomial is given by f(x) = ax^3 - x^2 + bx + 6. a) Find the values of the constants a and b. Step 1: Use the Factor Theorem. Since (x+2) is a factor of f(x), we know that f(-2) = 0. f(-2) = a(-2)^3 - (-2)^2 + b(-2) + 6 = 0 -8a - 4 - 2b + 6 = 0 -8a - 2b + 2 = 0 Dividing by -2, we get: 4a + b - 1 = 0 4a + b = 1 (*) Step 2: Use the Remainder Theorem. When f(x) is divided by (x+1), the remainder is 10. So, f(-1) = 10. f(-1) = a(-1)^3 - (-1)^2 + b(-1) + 6 = 10 -a - 1 - b + 6 = 10 -a - b + 5 = 10 -a - b = 5 (**) Step 3: Solve the system of linear equations. We have the system: 1. 4a + b = 1 2. -a - b = 5 Add equation (1) and equation (2): (4a + b) + (-a - b) = 1 + 5 3a = 6 a = 2 Substitute a=2 into equation (2): -(2) - b = 5 -2 - b = 5 -b = 7 b = -7 The values of the constants are a=2 and b=-7. b) Find the values of x for which f(x) = 0. Step 4: Write the complete polynomial and use the known factor. With a=2 and b=-7, the polynomial is f(x) = 2x^3 - x^2 - 7x + 6. Since (x+2) is a factor, x=-2 is a root. We can use synthetic division to find the other factors. c|cccc -2 & 2 & -1 & -7 & 6 \\ & & -4 & 10 & -6 \\ & 2 & -5 & 3 & 0 The quotient is 2x^2 - 5x + 3. So, f(x) = (x+2)(2x^2 - 5x + 3). Step 5: Find the