Well done to Edison, from STC School in Hong Kong, and to Esther, who both sent us correct solutions. Here is Esther's solution:

The first thing I did was to list all the combinations of +
and - signs that were possible and then I tried them out on two
sets of consecutive negative numbers.

I noticed that the answers to some of them were the same so I
tried to work out the general formulae using consecutive numbers
$n$, $n+1$, $n+2$, $n+3$.

The results I got are in the table below:

Operations | -8, -7, -6, -5 | -10, -9, -8, -7 | Formula |

+ + + | -26 | -34 | 4n + 6 |

+ + - | -16 | -20 | 2n |

+ - + | -14 | -18 | 2n + 2 |

+ - - | -4 | -4 | -4 |

- + + | -12 | -16 | 2n + 4 |

- + - | -2 | -2 | -2 |

- - + | 0 | 0 | 0 |

- - - | 10 | 14 | -2n - 6 |