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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