The given function is

Since k = 1 is a zero of f(x), that means
That means f(x) = 0 at x = 1
Let us find the factor from x = 1

The factor is (x - 1)
We will divide f(x) by (x - 1) to find the other factors

Multiply 2x^2 by (x -1 ) then subtract the answer from f(x)

Divide -x^2 by x, then multiply the answer by (x - 1), then subtract the result from the answer above

Divide -6x by x, then multiply the result by (x - 1), then subtract the answer from the last answer above

Then

Now we will factorize it into two factors

Then the 2 factors are (2x + 3) and (x - 2)
The factors of f(x) are
(x - 1) (x - 2) (2x + 3)