I want to prove the inequality $$\begin{aligned} &\sqrt{(x - 1)^2 + y^2}\Big[y^2(9x - 6) - 9x^2 + 9x^3\Big]+ y^2(16x^2 - 16x + 7)\\ &- \sqrt{x^2 + y^2}\Big[9x + y^2(9x - 3) + \sqrt{(x - 1)^2 + y^2}(9x^2 - 9x + 6y^2) - 18x^2 + 9x^3\Big]\\ & + 9x^2 - 18x^3 + 9x^4 + 7y^4 \gt 0 \end{aligned} $$ for real and nonzero $x,y$. Is this inequality true? How does one go about showing it?

I want to prove the inequality $$\begin{aligned} &\sqrt{(x - 1)^2 + y^2}\Big[y^2(9x - 6) - 9x^2 + 9x^3\Big]+ y^2(16x^2 - 16x + 7)\\ &- \sqrt{x^2 + y^2}\Big[9x + y^2(9x - 3) + \sqrt{(x - 1)^2 + y^2}(9x^2 - 9x + 6y^2) - 18x^2 + 9x^3\Big]\\ & + 9x^2 - 18x^3 + 9x^4 + 7y^4 \geq 0 \end{aligned} $$ for real and nonzero $x,y$. Is this inequality true? How does one go about showing it?

I want to prove the inequality $((x - 1)^2 + y^2)^{1/2}*(y^2*(9*x - 6) - 9*x^2 + 9*x^3) + y^2*(16*x^2 - 16*x + 7) - (x^2 + y^2)^{1/2}*(9*x + y^2*(9*x - 3) + ((x - 1)^2 + y^2)^{1/2}*(9*x^2 - 9*x + 6*y^2) - 18*x^2 + 9*x^3) + 9*x^2 - 18*x^3 + 9*x^4 + 7*y^4 > 0$ for real and nonzero $x,y$. Is this inequality true? How does one go about showing it?

I want to prove the inequality $$\begin{aligned} &\sqrt{(x - 1)^2 + y^2}\Big[y^2(9x - 6) - 9x^2 + 9x^3\Big]+ y^2(16x^2 - 16x + 7)\\ &- \sqrt{x^2 + y^2}\Big[9x + y^2(9x - 3) + \sqrt{(x - 1)^2 + y^2}(9x^2 - 9x + 6y^2) - 18x^2 + 9x^3\Big]\\ & + 9x^2 - 18x^3 + 9x^4 + 7y^4 \gt 0 \end{aligned} $$ for real and nonzero $x,y$. Is this inequality true? How does one go about showing it?