First, I think the polynomial should be:
The list of rational zeroes is based on the factors of the x³ coefficient 4=(1,4), (2,2) and the factors of the constant 9=(1,9), (3,3).
So we combine these possible factors to arrive at the given list of zeroes, which can be positive or negative.
For example, 3/2 combines (2,2) with (3,3), giving the rational zeroes ±3/2.
We then use all possible rational zeroes and synthetic division to test whether the rational zero is a true zero. When we get a true zero we derive a quadratic directly from the quotient of the division.
It would take too long and too much space to show all synthetic divisions so I’ll use just two examples to demonstrate the process.
First, let’s try -1 as a zero:
-1 | 4 19 -41 9
4 -4 -15 56
4 15 -56 | 65. This is the remainder, so -1 is not a true zero.
¼ | 4 19 -41 9
4 1 5 -9
4 20 -36 | 0. There’s no remainder, so ¼ is a true zero.
The quotient gives us the quadratic:
4x²+20x-36. So the cubic reduces to (x-¼)(4x²+20x-36).
This is the same as (4x-1)(x²+5x-9).
Therefore the quadratic is x²+5x-9.