1. **State the problem:** Factorise the quadratic expression $8x^2 + 22x + 15$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers that multiply to $a \times c$ and add to $b$. 3. **Calculate the product and sum:** Here, $a=8$, $b=22$, $c=15$. So, $a \times c = 8 \times 15 = 120$. 4. **Find two numbers that multiply to 120 and add to 22:** These numbers are 10 and 12 because $10 \times 12 = 120$ and $10 + 12 = 22$. 5. **Rewrite the middle term using these numbers:** $$8x^2 + 10x + 12x + 15$$ 6. **Group terms:** $$(8x^2 + 10x) + (12x + 15)$$ 7. **Factor each group:** $$2x(4x + 5) + 3(4x + 5)$$ 8. **Factor out the common binomial:** $$(2x + 3)(4x + 5)$$ **Final answer:** $$8x^2 + 22x + 15 = (2x + 3)(4x + 5)$$