I'm not quite sure what you mean by common core way but the way to get the answer the question is this:
1.) You must covert the whole numbers into fractions.
[tex]4 \frac{1}{5} \: will \: become \: \frac{21}{5} [/tex]
(multiply the denominator by the whole number, which in this case is 4, then add the answer to the numerator).
[tex]4 \frac{3}{8 \: } \: will \: become \: \frac{35}{8} [/tex]
(same steps will apply as before except in this case there is a different denominator and numerator).
There is no need to make the denominators the same number since it is multiplication and not addition or subtraction (the same can be done with division expect there is one crucial difference which I can explain if you want, in the comments).
2.) Next set up the equation to multiply:
[tex] \frac{21}{5} \times \frac{35}{8} [/tex]
5 and simplify (also note that the only way you can't simplify is across)
[tex] \frac{21}{5} \times \frac{7}{8} [/tex]
3.) Now multiply across:
NUMERATOR
21 × 7 = 147
DENOMINATOR
5 × 8 = 40
It will come together to be 147/40 or 3 27/40
