$\sin \alpha +\cos \alpha =\frac{5}{4}\mid {\left(\right)}^2$  

${\left(\sin \alpha +\cos \alpha \right)}^2={\left(\frac{5}{4}\right)}^2$  

${\sin }^2\alpha +2\sin \alpha \cos \alpha +{\cos }^2\alpha =\frac{25}{16}$     

$\underset{=1}{\underbrace{{\sin }^2\alpha +{\cos }^2\alpha }}+2\sin \alpha \cos \alpha =\frac{25}{16}$   

$1+2\sin \alpha \cos \alpha =\frac{25}{16}|-1$  

$2\sin \alpha \cos \alpha =\frac{9}{16}\mid :2$   

$\sin \alpha \cos \alpha =\frac{9}{32}$  

 

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${\left(\sin \alpha -\cos \alpha \right)}^2-3={\sin }^2\alpha -2\sin \alpha \cos \alpha +{\cos }^2\alpha -3=\underset{=1}{\underbrace{{\sin }^2\alpha +{\cos }^2\alpha }}-2\cdot \underset{\frac{9}{32}=}{\underbrace{\sin \alpha \cos \alpha }}-3=1-2\cdot \frac{9}{32}-3=-2-\frac{9}{16}=-2\frac{9}{16}$