1. **State the problem:** Fully factorise the quadratic expression $$3t^2 + 16t + 5$$. 2. **Recall the factoring method:** For a quadratic $$at^2 + bt + c$$, we look for two numbers that multiply to $$a \times c$$ and add to $$b$$. 3. **Calculate product and sum:** Here, $$a=3$$, $$b=16$$, and $$c=5$$. Calculate $$a \times c = 3 \times 5 = 15$$. We need two numbers that multiply to 15 and add to 16. 4. **Find the numbers:** The numbers are 15 and 1 because $$15 \times 1 = 15$$ and $$15 + 1 = 16$$. 5. **Rewrite the middle term:** Rewrite $$16t$$ as $$15t + 1t$$: $$3t^2 + 15t + 1t + 5$$ 6. **Group terms:** Group the terms in pairs: $$(3t^2 + 15t) + (1t + 5)$$ 7. **Factor each group:** $$3t(t + 5) + 1(t + 5)$$ 8. **Factor out the common binomial:** $$(3t + 1)(t + 5)$$ **Final answer:** $$3t^2 + 16t + 5 = (3t + 1)(t + 5)$$.