45
27
182
54
General term in the expansion is
tr+1 = (18Cr)(x13)18-r(12x-13)r = (18Cr)(12r)x13(18-r)-13r = (18Cr)(12r)x234 - 26r
For x², put 234 - 26r = 2, which gives 26r = 232, so r = 232/26 = 8
For x⁴, put 234 - 26r = 4, which gives 26r = 230, so r = 230/26 = 115/13 (not an integer, so there's a mistake in the question or solution)
Let's assume the question meant (x³ + 12x⁻³)¹⁸
Then the general term is tr+1 = (18Cr)(x³)18-r(12x⁻³)r = (18Cr)12rx3(18-r)-3r = (18Cr)12rx54 - 6r
For x², 54 - 6r = 2 => 6r = 52 => r = 52/6 (not an integer)
For x⁴, 54 - 6r = 4 => 6r = 50 => r = 50/6 (not an integer)
Let's assume the question meant (x³ + 12x⁻³)¹⁸. The general term is:
tr+1 = 18Cr (x3)18-r (12x-3)r = 18Cr 12r x54 - 6r
For x², 54 - 6r = 2 => r = 52/6 which is not an integer. This indicates an error in the problem statement.
For x⁴, 54 - 6r = 4 => r = 50/6 which is not an integer. This also indicates an error in the problem statement.
Let's reconsider the original problem (x¹³ + 12x⁻¹³)¹⁸. The general term is:
tr+1 = 18Cr (x13)18-r (12x-13)r = 18Cr 12r x234 - 26r
If we want the coefficient of x², then 234 - 26r = 2, which gives r = 232/26 which is not an integer.
If we want the coefficient of x⁴, then 234 - 26r = 4, which gives r = 230/26 which is not an integer.
There must be a mistake in the question. The provided solution is also incorrect because it uses r=12 and r=15 which do not yield x² and x⁴ terms.
The question is flawed, and no correct answer can be derived from the given information.