1. **State the problem:** We want to find which expressions are equivalent to $$15 f^4 - 60 f$$. 2. **Original expression:** $$15 f^4 - 60 f$$. 3. **Factor out the greatest common factor (GCF):** The GCF of $$15 f^4$$ and $$60 f$$ is $$15 f$$. $$15 f^4 - 60 f = 15 f (f^3 - 4)$$ 4. **Check each option:** - Option 1: $$15 f (f^3 - 4)$$ matches the factored form, so it is equivalent. - Option 2: $$5 (3 f^4 - 12 f) = 5 \times 3 f^4 - 5 \times 12 f = 15 f^4 - 60 f$$, so it is equivalent. - Option 3: $$3 f^4 (5 - 60 f) = 15 f^4 - 180 f^5$$, which is not equal to the original expression. - Option 4: $$3 f (5 f^3 - 20) = 15 f^4 - 60 f$$, so it is equivalent. - Option 5: $$15 (f^4 - 60 f) = 15 f^4 - 900 f$$, which is not equal to the original expression. - Option 6: $$5 f^3 (3 f - 12) = 15 f^4 - 60 f^3$$, which is not equal to the original expression. 5. **Summary:** The equivalent expressions are options 1, 2, and 4. **Note:** The user marked option 3 as equivalent, but it is not equivalent to the original expression.