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If under substitution you mean to entirely eliminate $y$, you can do so by computing resultant of your polynomial and $y^2-(x^3+x)$ with respect to $y$:

sage: R.<x,y> = PolynomialRing(QQ)                                                                                                                                                                                 
sage: (y^2+x*y+1).resultant(y^2-(x^3+x),y)                                                                                                                                                                         
x^6 - x^5 + 2*x^4 + x^3 + x^2 + 2*x + 1

If under substitution you mean to entirely eliminate $y$, you can do so by computing resultant of your polynomial and $y^2-(x^3+x)$ with respect to $y$:

sage: R.<x,y> = PolynomialRing(QQ) PolynomialRing(GF(43))                                                                                                                                                                                 
sage: (y^2+x*y+1).resultant(y^2-(x^3+x),y)                                                                                                                                                                         
x^6 - x^5 + 2*x^4 + x^3 + x^2 + 2*x + 1

If under substitution you mean It is also possible to entirely eliminate $y$, you can do so $y$ by computing resultant of your polynomial and $y^2-(x^3+x)$ with respect to $y$:

sage: R.<x,y> = PolynomialRing(GF(43))                                                                                                                                                                                 
sage: (y^2+x*y+1).resultant(y^2-(x^3+x),y)                                                                                                                                                                         
x^6 - x^5 + 2*x^4 + x^3 + x^2 + 2*x + 1

It If your polynomial is assumed to be zero, then it is also possible to entirely eliminate $y$ by computing resultant of your polynomial and $y^2-(x^3+x)$ with respect to $y$:

sage: R.<x,y> = PolynomialRing(GF(43))                                                                                                                                                                                 
sage: (y^2+x*y+1).resultant(y^2-(x^3+x),y)                                                                                                                                                                         
x^6 - x^5 + 2*x^4 + x^3 + x^2 + 2*x + 1