The equation of the line passing through point $ \left(-2,~5\right) $, and point $ \left(4,~-2\right) $ is:
$$ 7x+6y-16=0 $$To find equation of the line passing through points $ A(x_A,y_A) $ and $ B(x_B,y_B) $, we use formula:
$$ y - y_A~=~\frac{y_B - y_A}{x_B - x_A}(x-x_A) $$In this example we have:
$$ \begin{aligned} & \left(-2,~5\right) \implies x_A = -2 ~~\text{and}~~ y_A = 5 \\[1 em] & \left(4,~-2\right) \implies x_B = 4 ~~\text{and}~~ y_B = -2 \end{aligned} $$After substituting into the formula, we obtain:
$$ \begin{aligned} y - y_A~&=~\frac{y_B - y_A}{x_B - x_A}(x - x_A) \\[1 em] y - 5~&=~\frac{ -2 - 5 }{ 4 - \left(-2\right) } \left( x - \left(-2\right) \right) \\[1 em]y - 5 ~&=~ -\frac{ 7 }{ 6 } \left( x + 2 \right) \\[1 em]y - 5 ~&=~ -\frac{ 7 }{ 6 }x-\frac{ 7 }{ 3 } \\[1 em]y ~&=~ -\frac{ 7 }{ 6 }x + \frac{ 8 }{ 3 } \end{aligned} $$