Let's find the inflection points, we'll need to solve the equation for this
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative$$\frac{- \frac{2 x \left(2 x - \frac{x^{2} - 4}{x + 5}\right)}{x^{2} - 4} - \frac{4 x}{x + 5} + 2 + \frac{2 x - \frac{x^{2} - 4}{x + 5}}{x + 5} + \frac{2 \left(x^{2} - 4\right)}{\left(x + 5\right)^{2}}}{x^{2} - 4} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -5 - \frac{\sqrt{\frac{98}{\sqrt[3]{1470 \sqrt{2} + 2107}} + 2 \sqrt[3]{1470 \sqrt{2} + 2107} + 56}}{2} + \frac{\sqrt{- 2 \sqrt[3]{1470 \sqrt{2} + 2107} - \frac{98}{\sqrt[3]{1470 \sqrt{2} + 2107}} + \frac{840}{\sqrt{\frac{98}{\sqrt[3]{1470 \sqrt{2} + 2107}} + 2 \sqrt[3]{1470 \sqrt{2} + 2107} + 56}} + 112}}{2}$$
$$x_{2} = - \frac{\sqrt{- 2 \sqrt[3]{1470 \sqrt{2} + 2107} - \frac{98}{\sqrt[3]{1470 \sqrt{2} + 2107}} + \frac{840}{\sqrt{\frac{98}{\sqrt[3]{1470 \sqrt{2} + 2107}} + 2 \sqrt[3]{1470 \sqrt{2} + 2107} + 56}} + 112}}{2} - 5 - \frac{\sqrt{\frac{98}{\sqrt[3]{1470 \sqrt{2} + 2107}} + 2 \sqrt[3]{1470 \sqrt{2} + 2107} + 56}}{2}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -5$$
$$\lim_{x \to -5^-}\left(\frac{- \frac{2 x \left(2 x - \frac{x^{2} - 4}{x + 5}\right)}{x^{2} - 4} - \frac{4 x}{x + 5} + 2 + \frac{2 x - \frac{x^{2} - 4}{x + 5}}{x + 5} + \frac{2 \left(x^{2} - 4\right)}{\left(x + 5\right)^{2}}}{x^{2} - 4}\right) = \infty$$
$$\lim_{x \to -5^+}\left(\frac{- \frac{2 x \left(2 x - \frac{x^{2} - 4}{x + 5}\right)}{x^{2} - 4} - \frac{4 x}{x + 5} + 2 + \frac{2 x - \frac{x^{2} - 4}{x + 5}}{x + 5} + \frac{2 \left(x^{2} - 4\right)}{\left(x + 5\right)^{2}}}{x^{2} - 4}\right) = \infty$$
- limits are equal, then skip the corresponding point
Сonvexity and concavity intervals:Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left[- \frac{\sqrt{- 2 \sqrt[3]{1470 \sqrt{2} + 2107} - \frac{98}{\sqrt[3]{1470 \sqrt{2} + 2107}} + \frac{840}{\sqrt{\frac{98}{\sqrt[3]{1470 \sqrt{2} + 2107}} + 2 \sqrt[3]{1470 \sqrt{2} + 2107} + 56}} + 112}}{2} - 5 - \frac{\sqrt{\frac{98}{\sqrt[3]{1470 \sqrt{2} + 2107}} + 2 \sqrt[3]{1470 \sqrt{2} + 2107} + 56}}{2}, -5 - \frac{\sqrt{\frac{98}{\sqrt[3]{1470 \sqrt{2} + 2107}} + 2 \sqrt[3]{1470 \sqrt{2} + 2107} + 56}}{2} + \frac{\sqrt{- 2 \sqrt[3]{1470 \sqrt{2} + 2107} - \frac{98}{\sqrt[3]{1470 \sqrt{2} + 2107}} + \frac{840}{\sqrt{\frac{98}{\sqrt[3]{1470 \sqrt{2} + 2107}} + 2 \sqrt[3]{1470 \sqrt{2} + 2107} + 56}} + 112}}{2}\right]$$
Convex at the intervals
$$\left(-\infty, - \frac{\sqrt{- 2 \sqrt[3]{1470 \sqrt{2} + 2107} - \frac{98}{\sqrt[3]{1470 \sqrt{2} + 2107}} + \frac{840}{\sqrt{\frac{98}{\sqrt[3]{1470 \sqrt{2} + 2107}} + 2 \sqrt[3]{1470 \sqrt{2} + 2107} + 56}} + 112}}{2} - 5 - \frac{\sqrt{\frac{98}{\sqrt[3]{1470 \sqrt{2} + 2107}} + 2 \sqrt[3]{1470 \sqrt{2} + 2107} + 56}}{2}\right] \cup \left[-5 - \frac{\sqrt{\frac{98}{\sqrt[3]{1470 \sqrt{2} + 2107}} + 2 \sqrt[3]{1470 \sqrt{2} + 2107} + 56}}{2} + \frac{\sqrt{- 2 \sqrt[3]{1470 \sqrt{2} + 2107} - \frac{98}{\sqrt[3]{1470 \sqrt{2} + 2107}} + \frac{840}{\sqrt{\frac{98}{\sqrt[3]{1470 \sqrt{2} + 2107}} + 2 \sqrt[3]{1470 \sqrt{2} + 2107} + 56}} + 112}}{2}, \infty\right)$$