What number put into this equation as xwill give me a value of 0 for y?

No rational number will. It's an irrational number between -3 and -4.

A graphical solution can be quick and easy. The graph at the source link shows

.. x ≈ -3.26

Refinement can be accomplished by any of several means. Newton's iteration is quite fast at increasing the number of decimal places, giving

.. x ≈ -3.25992104989

after only two iterations starting from -3.26.

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The iterator can be written as

.. x' = (2/3)*((x +3)x^2 -5)/((x +4)x +4)

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Descartes' rule of signs tells you there are no positive real roots. It also tells you there are 1 or 3 negative real roots. The graph shows you there is only one negative real root. If there were a rational root, it would be a factor of 10, which is to say it would be -1, -2, or -5. The value is actually between -4 and -3, so is irrational.

y=2x^3+12x^2+24x+20=0=>

x=-3.259921

x=-1.37004+/-1.091124 i

approximately.