Final answer:
To find the value of m that will make the function g(x) have a remainder of 2 when divided by (x + 1), substitute x = -1 into the function and solve for m.
Step-by-step explanation:
To find the value of m that will make the function g(x) = mx² - 3x + x³ + 2 have a remainder of 2 when divided by the factor (x + 1), we can use the Remainder Theorem. The Remainder Theorem states that if you divide a polynomial by a linear factor (x - r), the remainder is equal to the value of the polynomial when x = r. In this case, the linear factor is (x + 1), so we need to find the value of m when x = -1.
Substituting x = -1 into the function, we get: g(-1) = m(-1)² - 3(-1) + (-1)³ + 2 = m + 3 + 1 + 2 = m + 6. Since the remainder is given as 2, we have m + 6 = 2. Solving for m, we subtract 6 from both sides: m = -4.