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Draw SM parallel  to TP  ....let M lie on QP

Draw TL perpendicular to SM......let L lie on SM

Draw QN perpendicular to TP....let QN intersect  SM  at O  and TP at N

In triangle  TLS ....since SM is  parallel to  TP....then  angle TLS = 90

And since TL is perpendicular to SM...angle LTP   =  90  so angle LTS =  120 - 90  = 30 

And angle TSL = 60

So triangle TSL =  30-60-90.... TS = 2 ,  LS  = 1  and TL  =sqrt (3)

And in triangle QOS,  angle QOS = 90

And angle OSQ = 120 -angle LST = 120 -60 = 60 

So triangle SOQ  also is 30- 60- 90

And  QS = 4  so OS= 2 and QO = 2sqrt 3

And  OS - LS  =  2 - 1   = 1  = LO =  TN

And NP =  PT -  TN =  5 - 1  =  4

So triangle QNP is a right triangle with  leg  NP = 4  and  leg QN = TL + QO =  3 sqrt 3  =  sqrt (27)

So  PQ  = sqrt [ 4^2 + (sqrt (27))^2  ]  = sqrt [ 16 + 27 ]  =  sqrt (43)