Explanation:

- We are given a triangle ABC with sides a, b, c and the perimeter p.

- The formula for the area (q) is derived using Heron’s formula:

$$q=\sqrt{s(s-a)(s-b)(s-c)}$$

where $s=\frac{p}{2}$, the semi-perimeter.

- The formula for $\tan (A/2)$ using the sides of the triangle is:

$$\tan (A/2)=\sqrt{\frac{(s-b)(s-c)}{s(s-a)}}$$

- Substituting into $p(p-2a)\tan (A/2)$, we simplify it to:

$$p(p-2a)\cdot \sqrt{\frac{(s-b)(s-c)}{s(s-a)}}=2\times 2q$$

- Therefore, the expression simplifies to $4q$.

So, the answer is Option 4: 4q.

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