Question

$y = \frac{x^{2} - 3x + c}{x^{2} + 3x + c}$

$$\begin{matrix} & \ \Rightarrow \ (y - 1)x^{2} + 3(y + 1)x + c(y - 1) = 0 \\ & D \geq 0\ \Rightarrow \ 9(y + 1)^{2} \geq 4c(y - 1)^{2} \\ & \text{~If~}c \leq 0\text{, then~}y \in R\ \Rightarrow \ c > 0 \\ & \ \Rightarrow ((3 + 2\sqrt{c})y - (2\sqrt{c} - 3))((2\sqrt{c} - 3)y - (3 + 2\sqrt{c})) \leq 0 \\ & \ \Rightarrow \ y \in \left\lbrack \frac{2\sqrt{c} - 3}{3 + 2\sqrt{c}},\frac{3 + 2\sqrt{c}}{2\sqrt{c} - 3} \right\rbrack \\ & \ \therefore\ \frac{3 + 2\sqrt{c}}{2\sqrt{c} - 3} = 7 \Rightarrow \sqrt{c} = 2 \Rightarrow c = 4 \end{matrix}$$