Pls see!,

It is not correct to write p(x) = q(x) only.

Instead of this it can be written p(x) = q(x) at x = 2, since p(x) and q(x) when divided by (x - 2) to obtain same remainder.

                    

So, p(2) = q(2).

That is 2 × 2³ + k × 2² - 3 × 2 + 5 = 2³ + 2 × 2² - 2 + k

Or,       2 × 8 + k × 4 - 6 + 5 = 8 + 2 × 4 - 2 + k

Or,       16 + 4k - 6 + 5 = 8 + 8 - 2 + k

Or,       15 + 4k = 14 + k

Or,       4k - k = 14 - 15

Or,       3k = -1

Or,       k = -1/3