If x>0x>0, we have x(x+4)=x^2+4xx(x+4)=x2+4x and (x+3)(x+1)=x^2+4x+3(x+3)(x+1)=x2+4x+3, we have
(x+3)(x+1) > x(x+4)(x+3)(x+1)>x(x+4)
i.e. (x+4)/(x+1) < sqrt2 < (x+3)/xx+4x+1<√2<x+3x
Hence x+4 < sqrt2x+sqrt2x+4<√2x+√2 i.e. x(sqrt2-1)>4-sqrt2x(√2−1)>4−√2
or x>(4-sqrt2)/(sqrt2-1)x>4−√2√2−1
and as (4-sqrt2)/(sqrt2-1)=(4-sqrt2)/(sqrt2-1)xx(sqrt2+1)/(sqrt2+1)4−√2√2−1=4−√2√2−1×√2+1√2+1
= 4-2+3sqrt2=2+4.2426=6.24264−2+3√2=2+4.2426=6.2426 i.e. x>6.2426x>6.2426 (A)
Further, as (x+3)/x > sqrt2x+3x>√2,
(sqrt2-1)x <3(√2−1)x<3 i.e. x < 3/(sqrt2-1)x<3√2−1
or x < 3sqrt2+3=7.2426x<3√2+3=7.2426 (B)
As such from (A) and (B), 6.2426 < x < 7.24266.2426<x<7.2426 and as xx is an integer x=7x=7
