5. Let the quadratic equation $x^{2}-3 x+a+4=0$ have two roots that are both integers. (1) Prove: (1) the roots must be one odd and one even. (2) $a$ is even. (2) When the roots have the same sign, find the roots and $a$.

Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
(Answer: $a=-2$ )