$$\large y=\left(x^2-2x+2\right) e^{\dfrac{5x}{2}}$$

using the product rule for derivatives

$$\frac{dy}{dx}=\frac{d}{dx} \left(x^2-2x+2\right) \cdot e^{\dfrac{5x}{2}}+\left(x^2-2x+2\right) \cdot \frac{d}{dx}\left(e^{ \dfrac{5x}{2}}\right) =$$

$$(2x-2)e^{\dfrac{5x}{2}}+\left(x^2-2x+2\right)\frac{5}{2}e^{ \dfrac{5x}{2}} =$$

$$e^{ \dfrac{5x}{2}}\left(\frac{5}{2}\left(x^2-2x+2\right)+(2x-2)\right)=$$

$$e^{ \dfrac{5x}{2}}\left(\frac{5}{2}x^2-3x+3\right)$$