Đáp án:

•  \( \overrightarrow{p} . \overrightarrow{q} \) \( = (\overrightarrow{a} - \overrightarrow{b}) . (\overrightarrow{a} + 2\overrightarrow{b}) = \overrightarrow{a}^2 - 2\overrightarrow{b}^2 + \overrightarrow{a}.\overrightarrow{b}\)

                 \(= 4-18 = -14\)

•  \(\overrightarrow{p}^2 = (\overrightarrow{a} - \overrightarrow{b})^2 = \overrightarrow{a}^2 - 2\overrightarrow{a} \overrightarrow{b} + \overrightarrow{b}^2 = 4 + 9 = 13 \quad \Rightarrow |\overrightarrow{p}| = \sqrt{13}
\)

•  \(\overrightarrow{q}^2 = (\overrightarrow{a} + 2\overrightarrow{b})^2 = \overrightarrow{a}^2 + 4\overrightarrow{b} + 4\overrightarrow{a} .\overrightarrow{b}^2 = 4 + 36 = 40 \quad \)

\(\Rightarrow |\overrightarrow{q}| = 2\sqrt{10}
\)

•  \(\overrightarrow{p}. \overrightarrow{q} = |\overrightarrow{p}| .|\overrightarrow{q}| \cos(\overrightarrow{p}, \overrightarrow{q})\)

\(\Rightarrow\cos(\overrightarrow{p}, \overrightarrow{q}) = \frac{\overrightarrow{p}. \overrightarrow{q}}{|\overrightarrow{p}| |\overrightarrow{q}|} = \frac{-14}{\sqrt{13}  .2\sqrt{10}} = -\frac{7\sqrt{130}}{130}
\)

\(\Rightarrow\) Vậy chọn \(\boxed{\text{A}} \)