Cos and Sin inside a Quadratic

The initial condition is that the $y$ -intercept is less than $0.25p^2$ , and $0.2p^2<0.25p^2$ for all values of $p$ . First, we can rewrite $\sin(x)+\cos(x) = \sqrt{2}*\sin{(x+45º)}$ using the R-Method. Let $x+45º=d$ . If $Q(x)=x^2-(a+b)x+(ab)=(x-a)(x-b)$ , then $Q(\sin{x}+\cos{x})$ has roots $a=\sqrt{2}\sin{d}$ and $b=\sqrt{2}\sin{d}$ . Since $\sin{d}$ is maximum at $1$ and minimum at $-1$ , then the maximum value of $p$ is $p=(a+b)=2\sqrt{2}$ and the minimum value is $p=(a+b)=-2\sqrt{2}$ , so the range length is $4\sqrt{2}$ and $(4\sqrt{2})^2=32$ which is the final answer.