we know that
g(x)------> has a root of multiplicity 2 at t=2 ------> (t-2)^2
roots of multiplicity 1 at t=0 and t=-3------> t(t+3)(t-2)^2
so
the equation of g(x) is
g(x)=at(t+3)(t-2)^2
where a is the leading coefficient
Remember that
g(5)=288
so
For x=5, g(x)=288
substitute the given values in the equation and solve for a
288=a(5)(5+3)(5-2)^2
288=a(5)(8)(3^2)
288=a(360)
a=288/360
a=0.8
therefore
the answer is
g(x)=0.8t(t+3)(t-2)^2