This problem shows the value and power of factor trees and prime factorisation for finding solutions to a problem that may at first glance seem almost impossible! A systematic approach is all that is needed...
This printable worksheet may be useful: Funny Factorisation
Draw the first product on the board, and explain that the missing cells contain the digits 1, 2, 5, 7 and 8 in some order so that the whole calculation uses each of the digits 1 to 9 once and only once:
Give students a bit of thinking time to make a plan for how they might solve the problem, and then bring them together to discuss approaches. Some might suggest working out which pairs of numbers multiply together to give a 6 in the units column. Others might think of finding all the factor pairs that multiply to give 4396 and looking for a pair that uses the appropriate digits. (If no-one suggests the second method, you could give them a hint that this might be a good approach, by drawing attention to the title of the problem.)
Students will need to think about how to combine the prime factors of 4396 to make 2-digit numbers, and then work systematically through the possibilities to find a pair that uses the available digits. Once they have tackled this first challenge, you can let them loose on the rest of the funny factorisations.
Does it help to draw a factor tree?
If you know the prime factors of a number, how can you work out which two-digit numbers are factors?
The "fill in the blanks" funny factorisation requires a different sort of systematic reasoning to complete.
A challenging extension for students who know a bit about computer programming might be to write a program that finds all seven funny factorisations.