Find $f$ such that $f^{\prime}(x)=8 x^{2}+5 x-2$ and $f(0)=2$

Step 1 ：We are given that the derivative of the function $f$ is $f^{\prime}(x)=8 x^{2}+5 x-2$ and that the function passes through the point $(0,2)$.

Step 2 ：To find the function $f$, we can integrate the derivative to find the antiderivative.

Step 3 ：The antiderivative of $f^{\prime}(x)=8 x^{2}+5 x-2$ is $f(x) = \frac{8}{3}x^3 + \frac{5}{2}x^2 - 2x + C$, where $C$ is the constant of integration.

Step 4 ：We can determine the value of $C$ by using the point $(0,2)$, which the function $f$ passes through.

Step 5 ：Substituting $x=0$ and $f(0)=2$ into the equation gives $C=2$.

Step 6 ：Substituting $C=2$ back into the equation gives the function $f(x) = \frac{8}{3}x^3 + \frac{5}{2}x^2 - 2x + 2$.

Step 7 ：\(\boxed{f(x) = \frac{8}{3}x^3 + \frac{5}{2}x^2 - 2x + 2}\) is the function such that $f^{\prime}(x)=8 x^{2}+5 x-2$ and $f(0)=2$.