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$$ (10a + b) - (6a + b) = 180 - 120 \Rightarrow 4a = 60 \Rightarrow a = 15 $$
Substitute $ a = 15 $ into (i):
$$ 6(15) + b = 120 \Rightarrow 90 + b = 120 \Rightarrow b = 30 $$
Substitute $ a = 15 $, $ b = 30 $ into (1):
$$ 4(15) + 2(30) + c = 120 \Rightarrow 60 + 60 + c = 120 \Rightarrow c = 0 $$
p(t) = 15t^2 + 30t
\boxed{15t^2 + 30t}
Question: A science fiction writer models the energy output $ E(t) $ of a fusion reactor on Mars as a cubic polynomial satisfying $ E(1) = 20 $, $ E(2) = 58 $, $ E(3) = 132 $, and $ E(4) = 26^3 $. Find $ E(0) $.
Solution: Let $ E(t) = at^3 + bt^2 + ct + d $. Use the given values:
$ E(1) = a + b + c + d = 20 $