Identify all intercepts of f(x)= x^2+7x+12/x^3+4x^2-x-4

Here is what you have written:

x^{2}+7x+\frac{12}{x^{3}}+4x^{2}-x-4

I assume you mean:

\frac{x^{2}+7x+12}{x^{3}+4x^{2}-x-4}

This is why grouping symbols are important.

To find the y-intercepts, set x=0

To find the x-intercepts, set y=0.

This simplifies down. Factor the top and bottom and cancel accordingly to simplify.

\frac{(x+3)(x+4)}{(x-1)(x+1)(x+4)}