$p(x, y)=a^2x_1y_1+ab x_2y_1+abx_1y_2 +b^2x_2y_2$, $x=(x_1,y_1),y=(y_1,y_2)\in\Bbb{R}^2$. what values of $a$ & $b$ does $p$ define an inner product?

Question

For $a, b\in\Bbb{R}$, let
$p(x, y) =a^2x_1y_1 + ab x_2y_1+abx_1y_2 +b^2x_2y_2$, $x=(x_1,x_2),y=(y_1,y_2)\in \Bbb{R} ^2$
For what values of $a$ and $b$ does $p:\Bbb{R} ^2 \times \Bbb{R} ^2 \rightarrow \Bbb{R} $ define an inner product?
(1)$a>0,b>0$
(2)$ab>0$
(3)$a=0,b=0$
(4) For no values of $a, b$. Try: We know for an inner product $<x, x>>0$ Hence $f(x, x) =a^2{x_1}^2+abx_2x_1+abx_1x_2+b^2{x_2}^2$ should be $>0$.Now if we choose $a=1$ and $b=-1$ then $p(x, x) ={x_1}^2+{x_2} ^2$ which is clearly $>0$. Hence the first three options are false. Am I correct? I also want to know if there is any way to prove it without taking examples.