Final answer:
The equation of the requested polynomial is (3/32)(x-4)^2(x-1)(x+2), which takes into account the given roots, their multiplicities, and the y-intercept.
Step-by-step explanation:
To write the equation for the polynomial with degree 4 given the roots and y-intercept, we can use the fact that a polynomial's roots tell us the factors of the polynomial. Since the root at x=4 has a multiplicity of 2, we have the factor (x-4)^2. For the roots x=1 and x=-2, each with a multiplicity of 1, we have the factors (x-1) and (x+2), respectively.
Thus, the polynomial in its factored form is k(x-4)^2(x-1)(x+2), where k is a constant that we must solve for using the y-intercept. When x=0, y=-3, so substituting these values into the polynomial gives us -3 = k(-4)^2(-1)(2), leading to -3 = 16k(-1)(2). Solving for k gives us k = 3/32.
Therefore, the full equation of the polynomial is (3/32)(x-4)^2(x-1)(x+2).