Firstly, we see that some tiles can combine to $2 \times 3$ tiles lying around shaded square. and there are $2$ ways arranging $2 \times 3$ tiles around shaded square.
In each case :
- in one $2 \times 3$ tile: there are $3$ ways arranging three $1 \times 2$ tiles so that we have $3^4=81$ for each way but some way to tile $5 \times 5$ grid can be counted in both cases. In that way , $2 \times 2$ tile in $4$ corners is tiled by two $1 \times 2$ tiles. Hence, there are $2^4=16$ ways. In conclusion , answer is $81 \times 2 - 16 = 146$ Am I wrong?