F(x)=F(1)+F'(1)(x-1)+F''(1)(x-1)^2/2!+F'''(1)(x-1)^3/3!+(x-1)^4(...)

F'(x)=fg(int_hk)+hk(int_fg)-fk(int_gh)-gh(int_fk)
F''(x)=(fg)'(int_hk)+(hk)'(int_fg)-(fk)'(int_gh)-(gh)'(int_fk)
F'''(x)=(fg)''(int_hk)+(hk)''(int_fg)-(fk)''(int_gh)-(gh)''(int_fk)

so by remainder theorem F(1)+F'(1)(x-1)+F''(1)(x-1)^2/2!+F'''(1)(x-1)^3/3! is divisible by (x-1)^4 (of course assuming f,g,h,k are all at least degree 1)

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