1. **State the problem:** Factorize the expression $25x^2 - 5x + 30x^3$ by grouping. 2. **Recall the method:** Factorization by grouping involves grouping terms with common factors and factoring each group separately. 3. **Group terms:** Group the first two terms and the last term separately: $$ (25x^2 - 5x) + 30x^3 $$ 4. **Factor out the greatest common factor (GCF) from each group:** - From $25x^2 - 5x$, the GCF is $5x$, so factor it out: $$ 5x(5x - 1) $$ - From $30x^3$, the GCF is $30x^3$, so factor it out: $$ 30x^3 $$ 5. **Rewrite the expression:** $$ 5x(5x - 1) + 30x^3 $$ 6. **Notice that the second term $30x^3$ can be rewritten to factor out $(5x - 1)$:** Rewrite $30x^3$ as $6x^2(5x - 1)$? Check if this is possible: Actually, $6x^2(5x - 1) = 30x^3 - 6x^2$, which is not equal to $30x^3$. So this approach does not work. 7. **Try rearranging terms for better grouping:** Group $25x^2$ and $30x^3$ together, and $-5x$ separately: $$ (25x^2 + 30x^3) - 5x $$ 8. **Factor out GCF from the first group:** $$ 5x^2(5 + 6x) - 5x $$ 9. **Factor out GCF from the entire expression:** Notice $5x$ is common in both terms: $$ 5x( x(5 + 6x) - 1 ) $$ 10. **Simplify inside the parentheses:** $$ 5x(5x + 6x^2 - 1) $$ 11. **Rewrite the expression:** $$ 5x(6x^2 + 5x - 1) $$ 12. **Factor the quadratic $6x^2 + 5x - 1$:** Find two numbers that multiply to $6 imes (-1) = -6$ and add to $5$. These numbers are $6$ and $-1$. Rewrite the middle term: $$ 6x^2 + 6x - x - 1 $$ Group terms: $$ (6x^2 + 6x) - (x + 1) $$ Factor each group: $$ 6x(x + 1) - 1(x + 1) $$ Factor out common binomial: $$ (x + 1)(6x - 1) $$ 13. **Final factorization:** $$ 5x(x + 1)(6x - 1) $$ **Answer:** The factorization of $25x^2 - 5x + 30x^3$ is $$5x(x + 1)(6x - 1)$$.