Multiplication and division of integers both work in a very similar pattern to adding and subtracting. The short description of the process is that we multiply or divide like normal, and if the signs match (both positive or both negative) the answer is positive. If the signs don't match (one positive and one negative), the answer is negative.
Here are examples of multiplication and division of integers with matching signs:

Signs match, answer is positive  

Our Solution  
 

Signs match, answer is positive  

Our Solution 
In each example, since the two integers have matching signs, we multiply or divide the two numbers, and write the product or quotient as a positive number.
Here are some examples of multiplication and division of integers with opposite signs:

Signs do not match, answer is negative  

Our Solution  
 

Signs do not match, answer is negative  

Our Solution 
In these examples, the two integers have opposite signs: one is positive and the other is negative. We multiply or divide the two numbers, and write the product or quotient as a negative numbers.
Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT: HTTP://WALLACE.CCFACULTY.ORG/BOOK/BOOK.HTML