Factors of $x^2+3\sqrt{2}x+4$ are

To factorize $x^2+3\sqrt{2}x+4$ by splitting the middle term, we find two numbers whose sum is $3\sqrt{2}$ and whose product is $4$.The numbers are $2\sqrt{2}$ and $\sqrt{2}$:
$$\text{Sum}=2\sqrt{2}+\sqrt{2}=3\sqrt{2}$$
$$\text{Product}=2\sqrt{2}\times \sqrt{2}=2\times 2=4$$
Rewrite the expression:
$$x^2+2\sqrt{2}x+\sqrt{2}x+4$$
$$x(x+2\sqrt{2})+\sqrt{2}(x+2\sqrt{2})$$
$$(x+2\sqrt{2})(x+\sqrt{2})$$