I actually just answered the exact same question for someone else.
Answer:
y(x)=-C(x-1)^3(x+4)(x+2)
Step-by-step explanation:
Look at each point the graph passes the x-axis.
At this line, y=0, we can make factors of the polynomial.
It passes through x=-4, x=-2, and x=-1
Since each factor is written as (x-k), we put the opposite,
so (x+4) for passing through x=-4
At x=1, it becomes more of a special case as it isn't a simple pass through,
but instead flattens a little. If it bounces back up, the exponent is even.
Since it passes through, we know the exponent is odd.
In this scale, I assumed the flat part is about 3.
Similarly, I cannot see the maxima and minima,
so there may be an additional constant missing to vertically elongate or flatten the graph, which I have input at 'C'.
Lastly, I can see that the polynomial is odd (largest power is 5 I believe), so since it begins and ends decreasing, the polynomial has a negative constant, hence the negative sign at the beginning, meaning C>0 as I accounted for the confirmed negative value.