Step 1: Recall what a longstanding multinodular goitre is.
A multinodular goitre (MNG) is an enlarged thyroid gland with many nodules, often from long term iodine deficiency or chronic stimulation of the gland. Over many years, a small number of these glands develop cancer within one of the nodules.
Step 2: Recall the malignancy linked with longstanding MNG.
Follicular carcinoma is the thyroid cancer classically described as arising in a background of longstanding multinodular goitre, especially in areas of iodine deficiency. The chronic overstimulation of follicular cells over years favors this pattern of malignant change.
Step 3: Contrast with the other thyroid cancers.
(A) Papillary carcinoma: This is the most common thyroid cancer overall and is linked mainly to radiation exposure, not to longstanding goitre. It usually arises in an otherwise normal gland.
(C) Anaplastic carcinoma: This is a rare, aggressive cancer seen mostly in elderly patients, and while it can rarely follow a longstanding goitre, it is not the classic association being tested here.
(D) Medullary carcinoma: This arises from the parafollicular C cells and is linked to MEN 2 syndromes, not to iodine deficient multinodular goitre.
Step 4: Final answer.
The malignancy classically linked with longstanding multinodular goitre is follicular carcinoma.
\[ \boxed{\text{Follicular carcinoma}} \]