The equation of the polynomial graphed is
f (x) = -1/48 (x + 3) (x + 2) (x - 2) (x - 4)²
How to find the equation
The equation of the polynomial is solved using the formula
f (x) = a (x - p) (x - q) (x - r).....
Where p, q, r, ... are the roots of the equation
From the graph the roots are x = -3, x = -2, x = 2 and x = 4 twice
In factored form we have
f (x) = a (x + 3) (x + 2) (x - 2) (x - 4)²
To solve for a we use point (0, 4)
4 = a (0 + 3) (0 + 2) (0 - 2) (0 - 4)²
4 = a 3 * 2 * -2 * 16
a = 4 / -192
a = -1 / 48
So the equation of the polynomial is
f (x) = -1/48 (x + 3) (x + 2) (x - 2) (x - 4)²