Roots of the quadratic equation $x^2+5x+3=0$ are $4\sin^2\alpha+a$ and $4\cos^2\alpha+a$. Another quadratic equation is $x^2+px+q=0$ where $p,q\in\mathbb{N}$ and $p,q\in[1,10]$. Find the probability that the roots of second quadratic equation are real and that they are $4sin^4\alpha+b$ and $4\cos^4\alpha+b$.
$$p^2-4q\ge 0$$
If $p=1$, then no possibilities.
If $p=2$, then $q=1$.
If $p=7,8,9,10$, then $q\in[1,10]$.
But in this way there may be repetitions. I need to find the number quadratic equations first and then I can use the fact the difference in roots for both equations is same to reduce the total possibilities.