Since the range is 10, the first and last scores must be different. And we only have 4 scores total. So in order for there to be a mode, and for that mode to be 44, it must be that the middle two scores are both 44 (just like you have already put 
)
Then let's let a be the first (lowest) score, and let d be the last (highest) score.
Since the mean is 45, we know:
(a + 44 + 44 + d) / 4 = 45
(a + 88 + d) / 4 = 45
Multiply both sides of the equation by 4
a + 88 + d = 180
Subtract 88 from both sides of the equation.
a + d = 92
Subtract d from both sides of the equation.
a = 92 - d
Since the range is 10, we know:
d - a = 10
Substitute 92 - d in for a
d - (92 - d) = 10
Distribute -1 to both terms in parenthesees
d - 92 + d = 10
Combine like terms
2d - 92 = 10
Add 92 to both sides
2d = 102
Divide both sides by 2
d = 51
Now we can find a using this value of d:
a = 92 - d
Substitute 51 in for d
a = 92 - 51
a = 41
