When multiplying monomials you need to first determine if they have the same base(s). For example, if you have x2 * x4 you can just keep the base (because they have the same base) and add the exponents together to get x6. This is because x2 really means x*x and x4 really means x*x*x*x. When these two are multiplied together you end up with x*x*x*x*x*x which is equal to x6.
When there are coefficients (numbers in front of the variable) involved you can just multiply the coefficients and then consider the bases as explained above. For example, 5y3(7y3). You can multiply the coefficients (the 5 and the 7) to get 35 and then multiply monomials with like bases (y3 and y3) to get y6 . The overall answer should be 35y6.
For the more complicated situation there could be more than one variable. In this case you can only combine like bases. For example: 4mn5(10m2n3). Multiply the coefficients first to get 40. Combine the "m" terms to get m3 (m=m1 so the exponent of 1 needs to be added to the exponent of 2 to get m3). Combine the "n" terms to get n8. The overall answer should be 40m3n8.
Hope this helps!