There are three odd number and two even numbers among 1,2,3,4,5.

The only way the left side could be odd would be for all three odds
to be on the left side.  Then the two evens would have to be on
the right making the right side even.  Since the left side cannot be odd,
neither can the right side. So both sides are even.

Since both evens, 2 and 4 cannot be on the right, and the right must be even,
then the two numbers on the right are odd.  So we have two evens and one odd
on the left and two odds on the right.

So we either have

EVEN×EVEN÷ODD = ODD + ODD  or

EVEN×ODD÷EVEN = ODD + ODD

Neither 3,4, nor 5 will divide evenly into the product of any
two of the other numbers, so the divided number on the left is either
1 or 2.  So we have either

EVEN×EVEN÷1 = ODD + ODD or
EVEN×ODD÷2 = ODD + ODD

The first case can only be

2×4÷1 = 3 + 5  which is a solution

The second case can only be

4×ODD÷2 = ODD + ODD

The added odd numbers on the
right are either 1+3, 1+5, or 3+5 or 8. So the right side
is either 4,6, or 8

The odd number on the left cannot be 1, since the right side cannot be
2.

The odd number on the left cannot be 5, since the right side cannot be
10.

So, the odd number on the left can only be 3, making the right side
3+5 or 8.

Therefore the two solutions are:

2×4÷1 = 3 + 5  and 3×4÷2 = 1 + 5

Edwin