did you mean the function is
h(t) = t³ - 4t² - 81t + 324
the rest of this answer bases on this.
the roller coaster enters the underground tunnel at 0 ft height, and leaves the tunnel again at 0 ft height.
so, we need to find the zeros of the given function.
the difference between the zeros is the time the coaster is in the tunnel.
h(t) = 0 = t³ - 4t² - 81t + 324
a polynomial of the 3rd degree has actually 3 zeros.
we try to do factorization, which identified the zeros : every factor has a value for t to turn 0 and bring therefore the whole multiplication result to 0.
so, we need 3 factors :
(t + a)(t + b)(t + c) = t³ - 4t² - 81t + 324
(t² + (a+b)t + ab)(t + c)
t³ + ct² + (a+b)t² + c(a+b)t + abt + abc
t³ + (a+b+c)t² + (ca+cb+ab)t + abc
t³ = t³ done
a + b + c = -4
ca + cb + ab = -81
abc = 324
now, we have to solve this.
out of
a + b + c = -4
we know that at least one of the 3 variables has to be negative.
out of
abc = 324
we know then that exactly 2 of them have to be negative and one positive.
the factors of 324 are
1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, and 324
the prime factors are
2, 2, 3, 3, 3, 3
when combining them they create all the factors.
we are looking for a combination of 3 factors.
and they must be relatively close to each other, so that their sum can be so small as -4.
with a little trial and error we find that
324 = 4 × 9 × 9
for example 3×3×36 or 2×2×81 represent factors with sizes that are too far apart to provide a sum of -4 with whatever signs we can assign to them.
4, 9, 9 can be added to -4 only if 4 is negative and one 9 is negative.
324 = -4×9×-9
so, (a, b, c) for our factorization is
(-9, -4, 9)
and the factorization is then
(t - 9)(t - 4)(t + 9) = t³ - 4t² - 81t + 324
and the zeros are correspondingly
t = 9, 4, -9
t = -9 does not make any sense in our context (there cannot be negative time, and it is outside the given interval of 0 <= t <= 10).
so, what is left as regular solutions : t = 9, 4
that means the coaster enters the tunnel after 4 seconds and leaves it again after 9 seconds.
therefore,
the coaster is in the tunnel for 9-4 = 5 seconds.