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F=first; S=second=F+2; T=third=F+4; L=fourth=largest=F+6
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F+S+T+L=120
F+(F+2)+(F+4)+(F+6)=120
4F+12=120
4F=108
F=27
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L=F+6=27+6=33
ANSWER: The largest integer is 33.
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CHECK:
F=27
S=F+2=27+2=29
T=F+4=27+4=31
L=33
.
F+S+T+L=120
27+29+31+33=120
120=120