so, a line that is parallel to 5x-2y = -12, will have the same exact slope as that equation's, so what is it anyway?
[tex]\bf 5x-2y=-12\implies 5x+12=2y\implies \cfrac{5x+12}{2}=y
\\\\\\
\cfrac{5x}{2}+\cfrac{12}{2}=y\implies \stackrel{slope}{\cfrac{5}{2}}x+6=y[/tex]
so, notice, from the slope-intercept form, it happens that the slope is 5/2, well, the line will have the same slope.
so we're looking for the equation of a line whose slope is 5/2 and runs through -2,3
[tex]\bf \begin{array}{ccccccccc}
&&x_1&&y_1\\
&&(~ -2 &,& 3~)
\end{array}
\\\\\\
% slope = m
slope = m\implies \cfrac{5}{2}
\\\\\\
% point-slope intercept
\stackrel{\textit{point-slope form}}{y- y_1= m(x- x_1)}\implies y-3=\cfrac{5}{2}[x-(-2)]
\\\\\\
y-3=\cfrac{5}{2}(x+2)\implies y-3=\cfrac{5}{2}x+5\implies y=\cfrac{5}{2}x+8[/tex]
