How To Solve 5/x=10/3x+4?

There are several ways to interpret your question. Using the standard order of operations, it should be interpreted like this.
5/x = (10/3)*x + 4
5 = (10/3)x² + 4x    (multiply both sides by x)
(10/3)x² + 4x - 5 = 0    (subtract 5 from both sides; swap sides)
x² + 1.2 x - 1.5 = 0    (multiply both sides by 3/10)
x = (-1.2 ±√(1.2² - 4(1)(-1.5)))/(2*1)    (use the quadratic formula)
x = (-1.2 ±√(1.44+6))/2
x = -.6 ±√1.86
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A common error is to omit parentheses around the denominator expression. Assuming that error was made here, another way to interpret your question is
5/x = 10/(3x+4)
5(3x+4) = 10x    (multiply both sides by x(3x+4))
15x + 20 = 10x    (multiply it out)
5x + 20 = 0    (subtract 10x from both sides)
x + 4 = 0    (divide both sides by 5)
x = -4    (subtract 4 from both sides)