Let's break down and solve the problem step by step.

(a) Express $2x^2 + px + q$ in the form $a(x + b)^2 + c$.

This part requires completing the square.

1. Start with the quadratic expression: $2x^2 + px + q$

2. Factor out the coefficient of $x^2$ (which is 2 in this case) from the first two terms: $2(x^2 + \frac{p}{2}x) + q$

3. To complete the square, take half the coefficient of $x$, square it, and then add and subtract it inside the bracket: $\frac{p}{2} \quad \text{(half of this is)} \quad \frac{p}{4}, \quad \left( \frac{p}{4} \right)^2 = \frac{p^2}{16}$ Add and subtract this inside the bracket: $2 \left( x^2 + \frac{p}{2}x + \frac{p^2}{16} - \frac{p^2}{16} \right) + q$ Simplify: [ 2 \left( \left( x + \