1. **State the problem:** Factorise fully the expression $$30pq^4 + 20p^2q^3 + 10q$$. 2. **Identify the greatest common factor (GCF):** Look at the coefficients 30, 20, and 10. The GCF of these numbers is 10. 3. **Look at the variables:** The terms have $p$, $p^2$, and no $p$ in the last term, so the lowest power of $p$ common to all terms is $p^0 = 1$ (no $p$ in the last term), so $p$ is not common to all terms. For $q$, the powers are $q^4$, $q^3$, and $q^1$. The lowest power is $q^1$, so $q$ is common to all terms. 4. **Extract the GCF:** The GCF is $10q$. 5. **Divide each term by the GCF:** $$\frac{30pq^4}{10q} = 3pq^3$$ $$\frac{20p^2q^3}{10q} = 2p^2q^2$$ $$\frac{10q}{10q} = 1$$ 6. **Write the factorised form:** $$10q(3pq^3 + 2p^2q^2 + 1)$$ 7. **Check if the expression inside the parentheses can be factorised further:** - $3pq^3 + 2p^2q^2 + 1$ has no common factors. - No obvious factorisation patterns (like difference of squares, perfect square trinomials, or factoring by grouping) apply. Therefore, the fully factorised form is: $$\boxed{10q(3pq^3 + 2p^2q^2 + 1)}$$