1. **Problem:** Fully factorise the quadratic expression $2x^2 + 12x + 18$. 2. **Formula and rules:** To factorise a quadratic expression of the form $ax^2 + bx + c$, first look for the greatest common factor (GCF) of all terms. Then, if possible, factorise the quadratic trinomial using methods such as splitting the middle term or using the quadratic formula. 3. **Step 1: Find the GCF of $2x^2$, $12x$, and $18$** The GCF is $2$. 4. **Step 2: Factor out the GCF:** $$2x^2 + 12x + 18 = 2(x^2 + 6x + 9)$$ 5. **Step 3: Factorise the quadratic inside the parentheses:** We look for two numbers that multiply to $9$ and add to $6$. These numbers are $3$ and $3$. 6. **Step 4: Write the factorised form:** $$2(x + 3)(x + 3) = 2(x + 3)^2$$ 7. **Answer:** The fully factorised form of $2x^2 + 12x + 18$ is $$\boxed{2(x + 3)^2}$$