VTMO 2019 — Grade 9 — final Question 1

$a$)Solve $x^{2} + 3 x \sqrt{x + 1} = \left(\frac{27}{4}\right) \left(x + 1\right)$. $b$)Find prime pairs $\left(p , q\right)$ for which $p^{2} + 5 pq + 4 q^{2}$ is a square.