Instead of trying to factor the cubic, substitute each answer choice into $f(x)=x^{3}-4x^{2}-7x+10$ and see which one makes the result equal 0.

When you substitute negative values, carefully compute $x^{3}$, $x^{2}$, and $-7x$ step by step so you do not lose or flip any minus signs.

In Desmos, type f(x)=x^3-4x^2-7x+10 to define the function.

On separate lines, type f(-5), f(-4), f(-1), and f(1). Look at the outputs and identify which input makes the output exactly 0; that input is the zero of the function and matches the correct answer choice.

A zero (or root) of a function is a value of $x$ that makes the function equal 0.

So we are looking for the value of $x$ among the choices that makes

equal 0, that is, we need $f(x)=0$.

Substitute $x=-5$ into $f(x)$.

Since $f(-5)=-180\ne 0$, $x=-5$ is not a zero of the function.

Substitute $x=-4$ into $f(x)$.

Since $f(-4)=-90\ne 0$, $x=-4$ is not a zero of the function.

Substitute $x=-1$ into $f(x)$.

Since $f(-1)=12\ne 0$, $x=-1$ is not a zero of the function.

Now substitute $x=1$ into $f(x)$.

Here $f(1)=0$, so $x=1$ is a zero of the function. Therefore, the correct answer is choice D, $1$.