The interval in which y=x^2 e^2 x is increasing is

$y=x^2 e^{2 x} ~~\Rightarrow y'=2 x e^{2 x}+2 x^2 e^{2 x}$

so $y'=0 ~~\Rightarrow e^{2 x}(2 x)(1+x)=0$

$\Rightarrow x=0,-1$

Using wavy curve method

so y' > 0 for x ∈ (-∞, -1) υ (0, ∞)

so y is increasing