Let's work through the problem step by step.

Part (a)

We are asked to express the quadratic equation $2x^2 + px + q$ in the form $a(x + b)^2 + c$, which is the completed square form.

We will complete the square for the expression $2x^2 + px + q$.

1. Factor out 2 from the first two terms:
   $2(x^2 + \frac{p}{2}x) + q$

2. Complete the square for the expression inside the parentheses:

   • Take half of the coefficient of $x$ inside the parentheses, which is $\frac{p}{2}$,