Final answer:
The anterior pituitary hormone responsible for stimulating the secretion of T3 and T4 metabolic hormones is the Thyroid Stimulating Hormone (TSH).
Step-by-step explanation:
The anterior pituitary hormone that promotes the secretion of the metabolic hormones T3 and T4 is the Thyroid Stimulating Hormone (TSH). TSH is responsible for stimulating the thyroid gland, leading to the release of thyroid hormones triiodothyronine (T3) and thyroxine (T4), which are crucial in regulating the body's basal metabolic rate. This secretion is part of a negative feedback system involving the hypothalamus, pituitary gland, and thyroid gland to maintain hormone levels within a desired range.