First, let's simplify the expression inside the parenthesis:

$$4\sqrt{a}-\frac{4a-2\sqrt{a}}{2+\sqrt{a}}$$

We can simplify the fraction by finding a common denominator:

$$4\sqrt{a}-\frac{2(2\sqrt{a}+\sqrt{a})}{2+\sqrt{a}}$$

$$4\sqrt{a}-\frac{2(3\sqrt{a})}{2+\sqrt{a}}$$

$$4\sqrt{a}-\frac{6\sqrt{a}}{2+\sqrt{a}}$$

Now, let's simplify the entire expression:

$$(4\sqrt{a}-\frac{6\sqrt{a}}{2+\sqrt{a}})÷\frac{1}{2+\sqrt{a}}$$

We can simplify the division by multiplying by the reciprocal of the divisor:

$$(4\sqrt{a}-\frac{6\sqrt{a}}{2+\sqrt{a}})\cdot (2+\sqrt{a})$$

$$4\sqrt{a}(2+\sqrt{a})-6\sqrt{a}$$

$$8\sqrt{a}+4a-6\sqrt{a}$$

$$2\sqrt{a}+4a$$

So, the final simplified expression is $$2\sqrt{a}+4a$$.