Final answer:
To find the possible values of k for which the expressions x - x - 2 and r + kr - 10x + 6 have a common factor, we factorize each expression and set their common factors equal to each other. Solving the resulting equation yields k = -2.
Step-by-step explanation:
The given expressions are x - x - 2 and r + kr - 10x + 6. To find the possible values of k for which these expressions have a common factor, we need to factorize each expression. Factoring x - x - 2 gives us (x - 2)(x + 1). Factoring r + kr - 10x + 6 gives us r(1 + k) - 2(5x - 3).
Now, we set the common factors of the two expressions equal to each other. (x - 2)(x + 1) = r(1 + k) - 2(5x - 3). Expanding this equation and simplifying gives us x² - x - 2 = (r - 10)x + (-2r + 9).
Comparing the coefficients of x on both sides, we get -1 = r - 10, which implies r = 9. Now, we substitute r = 9 into the equation and solve for k. -2 = -2r + 9(1 + k), which simplifies to -2 = -18 + 9k. Solving this equation gives us k = -2.