factor m^2+24m+144?

If the question is asking you to factorize m^2+24m+144?, then (m+12)(m+12) is correct.
If the question is asking you to solve m^2+24m+144=0, then m=-12,-12

your answer is right ..

it is (m+12)(m+12)

If meant to be an equation , then :-
m² + 24m + 144 = 0
(m + 12)(m + 12) = 0
m = - 12 (twice)

m^2 + 24m + 144
= m^2 + 12m + 12m + 144
= (m + 12)(m + 12)
= (m + 12)^2

It is a perfect square quadratic equation, so it will have only one root.

And yes, -12 is correct.