Use the formula,
[tex]\Delta x=v_it+\dfrac12at^2[/tex]
where [tex]\Delta x[/tex] is the cart's displacement (from the origin), [tex]v_i[/tex] is its initial speed, [tex]a[/tex] is its acceleration, and [tex]t[/tex] is time.
[tex]\Delta x=\left(2.0\dfrac{\rm m}{\rm s}\right)(5.0\,\mathrm s)+\dfrac12\left(1.6\dfrac{\rm m}{\mathrm s^2}\right)(5.0\,\mathrm s)^2[/tex]
[tex]\implies\boxed{\Delta x=30\,\mathrm m}[/tex]
Alternatively, since acceleration is constant, we have
[tex]\dfrac{v_f+v_i}2=\dfrac{\Delta x}t[/tex]
That is, we have these two equivalent expressions for average velocity, where [tex]v_f[/tex] is the cart's final velocity. Solve for [tex]\Delta x[/tex]:
[tex]\dfrac{10\frac{\rm m}{\rm s}+2.0\frac{\rm m}{\rm s}}2=\dfrac{\Delta x}{5.0\,\mathrm s}[/tex]
[tex]\implies\boxed{\Delta x=30\,\mathrm m}[/tex]
