3. For what value of $k$ does the equation $x^{2}-k x+k+3=0$ have two real roots, both of which are in the interval $(1,4)$?

When $k$ is what value, the equation $x^{2}-k x+k+3=0$ has two real roots, and both roots are within the interval $(1,4)$?
(Prompt: Example 4. Answer: $6 \leqslant k<\frac{19}{3}$ )