If exactly one root of the quadratic equation ${ x }^{ 2 }-(k+\frac { 11 }{ 3 } )x-({ k }^{ 2 }+k+1)=0$ lies in (0, 3) then which one of the following relation is correct

For exactly one root to lie in (0, 3), the product of the function values at the endpoints must be negative, f(0)*f(3) < 0. f(0) = -(k^2+k+1), f(3) = 9 - 3(k+11/3) - (k^2+k+1) = 9 - 3k - 11 - k^2 - k - 1 = -k^2 - 4k - 3. So (k^2+k+1)(k^2+4k+3) < 0. Since k^2+k+1 is always positive, we need k^2+4k+3 < 0, which is (k+1)(k+3) < 0, so -3 < k < -1.