Solve $LaTeX: \displaystyle 10 x^{2} + 31 x + 24=0$

This is an ac method factoring. The factors of $LaTeX: \displaystyle ac = 240$ that add up to $LaTeX: \displaystyle 31$ are $LaTeX: \displaystyle 15$ and $LaTeX: \displaystyle 16$ . Separating gives $LaTeX: \displaystyle (10 x^{2} + 15 x)+(16 x + 24)=0$ . This gives $LaTeX: \displaystyle (5 x + 8)(2 x + 3)=0$ . The solutions are $LaTeX: \displaystyle x = - \frac{8}{5}$ and $LaTeX: \displaystyle x = - \frac{3}{2}$