Answer:
{ -1, -3i,3i,4,-4}
Step-by-step explanation:
I'm going to try to get the Rational Root Theorem to work for us.
Since the coefficient of leading term is 1 we just need to look at the factors of the constant.
Possible rational zeros are going to be the factors of -144.
So here are some possible rational zeros: 1,2,3,4,6,8,9,12,16,18,24,36,48,72 and also the negative version of these numbers are numbers we must consider.
I'm going to see if -1 works.
(-1)^5+(-1)^4-7(-1)^3-7(-1)^2-144(-1)-144
-1 + 1 +7 -7 +144 -144=0
So -1 is a zero so x+1 is a factor. I'm going to use synthetic division to see what multiplies to x+1 that will me the initial polynomial expression we had.
-1 | 1 1 -7 -7 -144 -144
| -1 0 7 0 144
| ________ __________________
1 0 -7 0 -144 0
So the (x+1)(x^4-7x^2-144)=0
The cool thing is that other factor is a sort of quadratic in disguise. That is it becomes a quadratic if you let u=x^2. So let's do that.
u^2-7u-144=0
(u+9)(u-16)=0
u=-9 or u=16
So x^2=-9 or x^2=16.
Square rooting both sides gives us:

So the solution set is { -1, -3i,3i,4,-4}