X is charged with murder.

During its direct case, should the prosecution be allowed to introduce evidence that X has been convicted twice for assault, has a bad rep for violence and recently hit a woman on a bus?

The prosecution claims that X's violent character is an element of the crime of murder...

Since the prosecution is barred from introducing character evidence, in this case there is need to rely on the exception in FRE 404(b) and  having provided the required notice, introduce this evidence to prove motive, opportunity, intent or preparation etc.
I hope this helps!!

But, the landscape changes if your question arises where D first "opened the door."  In this case, prosecutor on direct can introduce reputation and opinion of D.  Thus, the repuation portion in your question is still admissible.

As direct, circumstantial evidence of intent, this would never be admissible except in a PMBR book. 

So, it depends on where you got the Q.  It is inadamissible per NCBE and Finz, and may be admissible in a PMBR jurisdiction.  BarBri is split.

Best of luck.

Here are the keys to this question.

Criminal Case - Direct Examination - conviction of bad acts - reputation- prosecution.

Here is a good general rule: The prosecution in a criminal case cannot introduce on direct examination either conviction of bad acts or reputation of the accused unless there is an exception in the rules.

1) It is not character evidence - shows motive or MO

2) Knowledge

3) impeachment

4) truthfulness/dishonesty

5) bais

My Barbri notes say this is not admissible as the defendant's character is NEVER an element of the crime (except entrapment)

I don't understand this as I thought under FRE 405,character evidence is addmissible for a purpose other than propensity (ie when that trait is itself an element)?

The answer is that the prosecutor is wrong. A violent character isn't an element of a crime. You have to think about every part of the question.

First off, propensity character traits are Never admissible in criminal cases unless defendant opens the door. There are three ways in which defendant can open the door 1)offering his own good relevant character; 2)attacking a character trait of the V; or 3) arguing self-defense in a homicide case. Second, the Prosecution is saying that they want to bring this in on direct evidence, which is not allowed. Can only bring in specific bad acts on cross examination. Hope it helps!