Answer
$\dfrac{3 \sqrt 5-12}{11}$
Work Step by Step
Rationalize the denominator by multiplying $\sqrt5-4$ to both the numerator and the denominator:
$\dfrac{-3}{\sqrt 5+4}=\dfrac{-3}{\sqrt 5+4} \cdot \dfrac{\sqrt 5-4}{\sqrt 5-4}$
Use distributive property in the numerator and special formula $(a+b)(a-b)=a^2-b^2$ in the denominator.
$=\dfrac{-3\cdot \sqrt 5+-(-3)\cdot(4)}{(\sqrt 5)^2-4^2}$
$=\dfrac{-3\sqrt 5+12}{(\sqrt 5)^2-16}$
Use the rule $(\sqrt{a})^2=a, a>0$ to obtain:
$=\dfrac{-3\sqrt 5+12}{5-16}$
$=\dfrac{-3\sqrt 5+12}{-11}$
Multiply $-1$ to both the numerator and the denominator:
$=\dfrac{-1(-3 \sqrt 5+12)}{-1(-11)}$
$=\dfrac{3 \sqrt 5-12}{11}$
Hence, the correct answer is $\frac{3 \sqrt 5-12}{11}$.
