To help you answer your question as to why Malcolm X did not simply ask the police and the government to protect Black people's rights, it is important to understand how long the struggle of Black people to be treated equally had already been going on and how frustrating this would have been. For example, it had been the Constitutional right of Black people to vote ever since the Fifteenth Amendment was passed in the late 1800s, yet in practice this was made as difficult as possible, particularly in the South. The Voting Rights Act, designed to stop Black people from being discriminated against when attempting to exercise their right to vote, only came into effect in 1965, which was the year of Malcom X's death. All throughout Malcom X's life, he would have experienced racism as an inherent part of the police and parts of the government. While the were officially supposed to protect the rights of all people, irrespective of their skin color, there were sadly many racist policemen and politicians who did not want to help Black people in their struggle for recognition and equality. Therefore, you could infer that Malcom X probably felt that asking the police or the government for help would be futile and would only lead to further humiliation. Also, any government decision is known to not happen overnight, as a lot of paperwork would have been involved; meetings would have to be held and laws would have to be passed, which is a longwinded process even without the added difficulty of institutional racism.
Instead, Malcom X argues that Black people should stand up for their rights and use force when necessary, as he felt that this would be a much more effective and powerful way of fighting for their rights than waiting for the slow wheels of bureaucracy to turn.