Sum, Difference, and Multiple-Angle Formulas · Superchallenge problem

The numbers $\tan\alpha$ and $\tan\beta$ are the two roots of $x^2 - 5x + 2 = 0$. Let $\varphi = \alpha + \beta$. Find the value of $\sin^2\varphi - 5\sin\varphi\cos\varphi + 2\cos^2\varphi$.