Question 312699
Graphing the function, it looks like there are no real solutions.
The function never crosses the x-axis(=0).
{{{ graph( 300, 300, -10, 10, -5, 15,x^2+x+sqrt(x^2+x-2)) }}} 
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Algebraically,
{{{x^2+x+sqrt(x^2+x-2)=0}}}
{{{sqrt(x^2+x-2)=-x^2-x}}}
{{{x^2+x-2=x^4+2x^3+x^2}}}
{{{x^4+2x^3-x+2=0}}}
{{{ graph( 300, 300, -10, 10, -10, 10, x^4+2x^3-x+2) }}} 
No real solutions.