$\sqrt{180}·\sqrt{5}$
= $\sqrt{5·36}·\sqrt{5}$
= $\sqrt{36}·\sqrt{5}·\sqrt{5}$
= $6·5$
= $30$

$\sqrt{20}·\sqrt{\frac{4}{5}}$
= $\sqrt{20·\frac{4}{5}}$
= $\sqrt{4·4}$
= $\sqrt{16}$
= $4$

$\sqrt{3}·\left(\sqrt{3}+\sqrt{48}\right)$
= $\sqrt{3}·\sqrt{3}+\sqrt{3}·\sqrt{48}$
= $3+\sqrt{3}·\sqrt{3·16}$
= $3+\sqrt{3}·\sqrt{3}·\sqrt{16}$
= $3+3·4$
= $3+12$
= $15$

Nach der 1. biniomischen Formel ((a+b)²=a²+2ab+b²) gilt:

${\left(\sqrt{2}+\sqrt{18}\right)}^2$ = ${\left(\sqrt{2}\right)}^2+2·\sqrt{2}·\sqrt{18}+{\left(\sqrt{18}\right)}^2$
= $2+2·\sqrt{2}·\sqrt{2·9}+18$
= $2+2·\sqrt{2}·\sqrt{2}·\sqrt{9}+18$
= $2+2·2·3+18$
= $2+12+18$
= $32$

$\sqrt{10x}·\sqrt{250x}$
= $\sqrt{10}·\sqrt{x}·\sqrt{250}·\sqrt{x}$
= $\sqrt{10}·\sqrt{250}·\sqrt{x}·\sqrt{x}$
= $\sqrt{10}·\sqrt{10·25}·x$
= $\sqrt{10}·\sqrt{10}·\sqrt{25}·x$
= $10·5·x$
= $50x$

Wir können die $\sqrt{31}$ wie Einheiten behandeln und einfach die $\sqrt{31}$ verrechnen.

$-14\sqrt{31}-9\sqrt{31}$
= $-23\sqrt{31}$