Question 405010
Here is the graph of :


{{{ y = 5x^2 + 4x + 6}}}


{{{ graph( 500, 500, -10, 10, -10, 10, 5x^2 + 4x + 6) }}}



Notice that the equation {{{5x^2 + 4x + 6}}} can be written as {{{0 = 5x^2 + 4x + 6}}} which amounts to replacing y in {{{y = 5x^2 + 4x + 6}}} by 0. 


So we really have this system: 

{{{system(y = 5x^2 + 4x + 6,y=0)}}} 

And {{{y=0}}} is the equation of the {{{x-axis}}}. 

Therefore we look to see {{{what}}}{{{ points}}} if any the graph of {{{y = 45x^2 + 4x + 6}}} has in common with the {{{x-axis}}}. 

It appears that there is {{{no}}} point the curve has in common with the {{{x-axis}}} and it is not a tangent to the x-axis anywhere. 

So the equation {{{5x^2 + 4x + 6}}} appears to have no real number solution.