Can You Find The Numbers That Each Letter Stands For In The Problem Below. EFGH X4 = HGFE?

Based on the thousands digit, we know E<3. Based on the units digit, we know H and E are not zero, because they must be different. So E must be 1 or 2, but E cannot be 1 because there are no multiples of 4 that end in 1.

E = 2, H = 8 or 9, and the carry into the 10s digit must be 3.

Now we need two numbers F, G such that
4G+3 = F+10n  (n=0,1,2,3 because the largest multiple of a single digit and 4 is 36, which means n<=3)
4F+n = G or G+10

Solve the first for F and plug into the second:
4(4G+3-10n)+n = G or G+10
16G+12-39n = G or G+10
15G + 12 = 39n or 39n+10
15G = 39n - 12 or 39n - 2
(n=0) 15G = -12 or -2  (doesn't work)
(n=1) 15G = 27 or 37  (doesn't work)
(n=2) 15G = 66 or 56 (doesn't work)
(n=3) 15G = 105 or 115 (works for 105, which means there is no carry into the thousands digit, and H = 8))
15G = 105, so G = 7
F = 4G+3-10n = 4G-27 = 28-27
F = 1

Check:
2178 * 4 = 8712.