1. **State the problem:** Fully factorise the quadratic expression $2x^2 + 13x + 15$. 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, $a=2$, $b=13$, $c=15$. So, $a \times c = 2 \times 15 = 30$. 4. **Find two numbers that multiply to 30 and add to 13:** These numbers are 10 and 3. 5. **Rewrite the middle term using these numbers:** $2x^2 + 10x + 3x + 15$. 6. **Group terms:** $(2x^2 + 10x) + (3x + 15)$. 7. **Factor each group:** $2x(x + 5) + 3(x + 5)$. 8. **Factor out the common binomial:** $(x + 5)(2x + 3)$. **Final answer:** $$2x^2 + 13x + 15 = (x + 5)(2x + 3)$$