Functional Equation Polynomial

Let $t = x^2 - 1 \implies x^2 = t + 1$. Substituting into the first function: $f(t) = (t+1)^2 - 7(t+1) + k_1 = t^2 + 2t + 1 - 7t - 7 + k_1 = t^2 - 5t + k_1 - 6$

Now consider the second function with $u = x^3 - 2 \implies x^3 = u + 2$. Substituting into the second function: $f(u) = (u+2)^2 - 9(u+2) + k_2 = u^2 + 4u + 4 - 9u - 18 + k_2 = u^2 - 5u + k_2 - 14$

Since $f$ is the same function in both cases: $k_1 - 6 = k_2 - 14 \implies k_2 - k_1 = 14 - 6 = 8$