Factor $20=2^{2}\times 5$. Rewrite the square root of the product $\sqrt{2^{2}\times 5}$ as the product of square roots $\sqrt{2^{2}}\sqrt{5}$. Take the square root of $2^{2}$.
$$\frac{19}{2\sqrt{5}+1}+4-2\sqrt{5}$$
Rationalize the denominator of $\frac{19}{2\sqrt{5}+1}$ by multiplying numerator and denominator by $2\sqrt{5}-1$.
Consider $\left(2\sqrt{5}+1\right)\left(2\sqrt{5}-1\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.
