How to factor this polynomial?

As @Fernis pointed out in the comments,

You cannot factor out the $(x+20)$ as you have done. There is no common factor of $(x+20)$ between $x^2+x−16$ and $20$.

Using the Rational Root Theorem, you can know that the possible rational roots are $\pm 1, \pm2, \pm4, \pm5, \pm10, \pm20$.

Through inspection and polynomial/synthetic division, you can get $(x-2)^2(x-5)$, as @saulspatz said. Therefore, (d) is your answer.

The easiest to try is (d), because it says that the polynomial has a double root. We will look for a root of the derivative and check if it cancels the polynomial.

$$3x^2+2x-16=0\iff x=2\text{ or }x=-\dfrac83.$$

Now $p(2)=0$, bingo !