Thanks. I've never studied this topic specifically, so if someone else already knows it, I hope they'll answer you. I'll just continue clarifying details for the moment. I found the topic in the last chapter of a Discrete Math book I have, so I'll go by that, since you didn't fully explain the details.
The problem is,
Given two languages L and M prove that: L*M U M= L*M
I had assumed * was a binary operation, from the way this is written; but I see it is a unary operation, the Kleene closure, namely the language that consists of all finite concatenations of strings (words?) in a language. The binary operation indicated by juxtaposing language names is "concatenation", which you didn't mention; it's the language consisting of concatenations of strings in each language (in order). So "L*M" is actually L* M, the concatenation of the Kleene closure of L with M. So I would start by thinking through what this means, and then what it means to take the union of this with M itself.
(I'll admit that I could have just looked this up, but wanted to give you the opportunity to practice explaining it to someone else, which is important in learning something well, and might even have led to your answering your own question.)
Now, you said,
The first statement is true, and I presume you have theorems that support it; the second superficially looks plausible, but you haven't said why you think it is true. Again, do you have any theorems about how the union of languages works? Or are you applying a theorem about union of sets? You'll just need to support your claim by some sort of reasoning. It will probably not require much, but it does require something!
(You could use "[ MATH ] \epsilon [ MATH ]", with the spaces removed, to insert an epsilon, or just paste it in from another source.)