The set $H_a$ is a Convex set

Question

Let $x \in \mathbb{R}^3$ and $$p(x)=x_1^2+x_2^2+x_3^2-x_1x_2-x_2x_3-x_1x_3. $$ For any $a \in \mathbb{R}^3$, show that $$H_a={x\in \mathbb{R}^3: p(x) +a \cdot x+1 <0}$$ convex.

Please help me to solve the above problem. Thank you.

Answer 1

Hint

Show that $p(x)+a\cdot x+1$ is convex by finding its Hessian matrix and showing it to be positive semi-definite.