1. **State the problem:** Factor the trinomial $$5x^{2} + 31x + 30$$. 2. **Recall the factoring formula:** For a quadratic trinomial $$ax^{2} + bx + c$$, we look for two numbers that multiply to $$a \times c$$ and add to $$b$$. 3. **Calculate the product and sum:** Here, $$a = 5$$, $$b = 31$$, and $$c = 30$$. Calculate $$a \times c = 5 \times 30 = 150$$. We need two numbers that multiply to 150 and add to 31. 4. **Find the pair:** The numbers 25 and 6 satisfy this because $$25 \times 6 = 150$$ and $$25 + 6 = 31$$. 5. **Rewrite the middle term:** Rewrite $$31x$$ as $$25x + 6x$$: $$5x^{2} + 25x + 6x + 30$$ 6. **Group terms:** Group the terms in pairs: $$(5x^{2} + 25x) + (6x + 30)$$ 7. **Factor each group:** $$5x(x + 5) + 6(x + 5)$$ 8. **Factor out the common binomial:** $$(5x + 6)(x + 5)$$ **Final answer:** $$5x^{2} + 31x + 30 = (5x + 6)(x + 5)$$.