[tex]6m^2 + 5m - 3 = 0[/tex]
[tex]6m^2 + 5m = 3[/tex]
Divide both sides by 6:
[tex]m^2 + (5/6) m = 1/2[/tex]
Add (5/12)^2 to both sides.
[tex]m^2 + (5/6) m + (5/12)^2 = 1/2 + (5/12)^2 [/tex]
We engineered the left side to be a square. Simplify the right:
[tex](m + 5/12)^2 = 97/144[/tex]
Take square roots, remembering the plus/minus:
[tex]m + 5/12 = \pm \sqrt{97/144}[/tex]
97 has no square factors,
[tex]m + 5/12 = \pm \sqrt{97}/12[/tex]
[tex]m = \dfrac{-5 \pm \sqrt{97}}{12}[/tex]
That's the answer; how you fit it into the boxes I'll leave to you.
Check:
Let's check with the quadratic formula:
[tex]m = \dfrac{-5 \pm \sqrt{5^2 - 4(6)(-3)}}{12} = \dfrac{-5 \pm \sqrt{97}}{12} \quad\checkmark[/tex]
