Let ( n^2 + 5n + 6 = k^2 ) [ n^2 + 5n + (6 - k^2) = 0 ] Discriminant ( \Delta = 25 - 4(6 - k^2) = 25 - 24 + 4k^2 = 1 + 4k^2 ) must be perfect square. Let ( 1 + 4k^2 = m^2 \implies m^2 - 4k^2 = 1 \implies (m-2k)(m+2k)=1 ) Only integer solution: ( m=1, k=0 ) → then ( n^2+5n+6=0 \implies n=-2,-3 ) Check: ( n=-2 ): ( 4-10+6=0 ) perfect square? 0 is square ✅ ( n=-3 ): ( 9-15+6=0 ) ✅
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