Force on the object is $F=k-\beta t^2$

Acceleration of the particle,

$a=\frac{F}{m}=\frac{k-\beta t^2}{m}$

Acceleration is zero when $k=\beta t^2$ or $t^2=\frac{k}{\beta }$ or $t^2=\frac{8}{2}\Rightarrow t=2 \mathrm{s}$

Now, $a=\frac{dv}{dt}=\frac{k-\beta t^2}{m}\Rightarrow dv=\frac{k-\beta t^2}{m}·dt$

Integrating between given limits, we have 

$\Rightarrow {\int }_{v=-15}^{v\left(t=2 \mathrm{s}\right)}dv={\int }_{t=-2 \mathrm{s}}^{t=2 \mathrm{s}}\frac{k-\beta t^2}{m}dt$

$=\frac{3}{2}{\int }_{t=-2}^{t=2}\left(8-2t^2\right)dt$

$v\left(t=2 \mathrm{s}\right)-\left(-15\right)=\frac{3}{2}{\left[8t-\frac{2t^3}{3}\right]}_{-2}^2$

$\Rightarrow v_{\mathrm{at} t=2\mathrm{s}}=-15+\frac{3}{2}\times \left.8\left(2-\left(-2\right)\right)-\frac{2}{3}\left(8-{\left(-2\right)}^3\right)\right]$

$\Rightarrow v_{\mathrm{at} t=2\mathrm{s}}=-15+\frac{3}{2}\left(32-\frac{32}{3}\right)$

$=-15+\frac{3}{2}\left(\frac{64}{3}\right)$

$=-15+32=17 {\mathrm{ms}}^{-1}$