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2 solutions
Moderator note:
This solution has been marked incomplete. Why did you remove the absolute signs?
The idea is to determine if the total value in the absolute lines are positive or negative.
Actually, I'm not really sure what to do if it's a negative, but if it's positive then we can confidently remove the modulus from the algebra.
First, factorise the 2a^3-3a^2-2a from the algebra in the first modulus in your LHS. Then use a=2009 to confirm that the LHS modulus is positive since a>0, and a>2, and that (2a+1)>0. Adding this 2a^3-3a^2-2a with 1 will give a postive value. Thus the modulus sign on the LHS algebra can be removed.
Now, factorise 2a^3-3a^2-4a on the RHS modulus. You can use the quadratic equation root formula. Then repeat the method in the first step to confirm that the RHS is positive, so the modulus can be removed.
Now, we consider the original form of the algebra (the one not factorised), but this time without the modulus.
Simplifying the algebra yields : 1+2a=4019.