Having difficulty to solve this determinents.

Here is an alternative way:
[math]\det \left(\begin{array}{ccc} 2a & 2b & a+b \\ b+c & a-c & a \\ b-c & a+c & b \end{array} \right) = \det \left(\begin{array}{ccc} 2a & 2b & a+b \\ 2b & 2a & a+b \\ b-c & a+c & b \end{array} \right) = \det \left(\begin{array}{ccc} 2a & 2b & 0 \\ 2b & 2a & 0 \\ b-c & a+c & \frac{b-a}{2} \end{array} \right)[/math]
 
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