both of your equations can be factored
x^4-7x^2-144 =(x^2-16)(x^2 +9) =(x-4)(x+4)(x^2+9)
giving you 4 and -4 as real roots
the second one factors into (x^2-25)(x^2+1) =(x-5)(x+5)(x^2+1)
giving you 5 and -5
when you have only x^4, x^2 and a constant, you can factor as above just like
ax^2 +bx+c
The rational root theorem is a much longer process
you would have to try all the positive and negative factors (using synthetic division) of 144 to try and find rational roots