Complete the square.

Solve for x:
3 x^2 + 5 x = 0

Divide both sides by 3:
x^2 + (5 x)/3 = 0

Add 25/36 to both sides:
x^2 + (5 x)/3 + 25/36 = 25/36

Write the left hand side as a square:
(x + 5/6)^2 = 25/36

Take the square root of both sides:
x + 5/6 = 5/6 or x + 5/6 = -5/6

Subtract 5/6 from both sides:
x = 0 or x + 5/6 = -5/6

Subtract 5/6 from both sides:
Answer: | x = 0     or     x = -5/3

3x^2 + 5x 

= 3(x^2 + 5x/3)

= 3(x^2 + 5x/3 + ((5/3)/2)^2 - ((5/3)/2)^2)

= 3(x^2 + 5x/3 + 25/36 - 25/36)

= 3(x+5/6)^2 - 25/12.