Inequality for $|y(y+x)e^{-x^2}| $ where $x \geqslant |y|$

Question

My textbook says

$$|y(y+x)e^{-x^2}| \leq 2x^2e^{-x^2} $$ where $x \geqslant |y|$

I get $$|y(y+x)e^{-x^2}| = |e^{-x^2}(y^2+yx)|= e^{-x^2}|y^2+yx| $$. From here I am thinking that since I know that $x \geqslant y $ then the term $(x^2+x*x)$ is bigger than $(y^2+yx)$

Is that correct thinking? I always find these inequalities a bit tricky.

Answer 1

We need to prove that $$|y(y+x)|\leq2x^2,$$ which is true because $$|y(y+x)|\leq|y|(|y|+|x|)\leq2|x|^2=2x^2,$$ where the last inequality it's $$(x-|y|)(2x+|y|)\geq0.$$