As the last question of the fill-in-the-blank question, this question is difficult and involves comprehensive knowledge such as exponent, piecewise function, abstract function, inequality, sequence, function and equation.

But when solving, paying attention to methods, especially as the type of judging right or wrong, actually reduces the difficulty to some extent. On the other hand, it is more difficult because of multiple choices.

We are required to give full consideration to:

1, f(x)>0 means 2 (-x- 1)-3 > 0 means 2 (-x- 1) > 3 so-x-1>; The answer to logarithm (2)3 is correct.

2. Because f (x) >; -3, the original equation has a real number solution, which means that the equation x 2+8x-m = 0 has a real number solution greater than -3, that is, the discriminant >; =0 and f (-3) < 0

So the answer is wrong.

3. Obviously, g (1) = g (0) = f (-1)+2 = 0 can't be a geometric series.

4. Substitution value can test that the sum of all solutions of g(x)=2x satisfies the condition (special value method).

So the correct answers are 1 and 4.

Answer it! Looking forward to adoption.