ANSWER
The polynomial is:
[tex]x^4-5x^3-42x^2+104x+320[/tex]EXPLANATION
Note: If n is a zero of a polynomial, then (x-n) is a factor of the same polynomial. Hence, the factors for the given zeros are:
P(x) = (x+5)(x+2)(x-4)(x-8)
Let's multiply these out to obtain the polynomial in Standard Form.
[tex]\begin{gathered} P\mleft(x\mright)=(x+5)\mleft(x+2\mright)\mleft(x-4\mright)\mleft(x-8\mright) \\ P(x)=(x^2+2x+5x+10)(x^2-8x-4x+32) \\ P(x)\text{ = }(x^2+7x+10)(x^2-12x+32) \\ P(x)=x^4-12x^3+32x^2+7x^3-84x^2+224x+10x^2-120x+320 \\ P(x)\text{ = }x^4-5x^3-42x^2+104x+320 \end{gathered}[/tex]