Find a constant a such that the vector field
$$\mathbf{F}(x,y) = {3\cdot x\cdot \left(2\cdot y+1\right)}\mathbf{i}+{a\cdot x^2}\mathbf{j}$$
is conservative and find a scalar potential $\varphi$ for the vector field F such that $\varphi(0,0)=5$.
$$F_1=3x(2y+1)$$
$$F_2=ax^2$$
$$\frac{\delta F_1}{\delta y}=\frac{\delta F_2}{\delta x}$$
$$6x=2ax$$
so $a=3$ that is right. But how do I find a scalar potential $\varphi$ for the vector field F such that $\varphi(0,0)=5$?
Hint. You want a function $\phi(x,y)$ so that $\nabla\phi=\mathbf F$ so that this function becomes $5$ at the origin.
From these conditions you find two differential equations for $\phi,$ and a way to determine the arbitrary constant.