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Problems:

Factor by grouping:

1. $5n^3 - 10n^2 + 3n - 6$
2. $35xy - 5x - 56y + 8$

Factor the Trinomials:

3. $x^2 + 15x + 50$
4. $c^2 + 2c - 24$
5. $9k^2 + 66k + 21$
6. $2n^2 + 5n + 2$

Solutions:

1. $5n^3 - 10n^2 + 3n - 6$

Group terms: $(5n^3 - 10n^2) + (3n - 6)$ Factor each group: $5n^2(n - 2) + 3(n - 2)$ Factor common term: $(5n^2 + 3)(n - 2)$

2. $35xy - 5x - 56y + 8$

Group terms: $(35xy - 5x) + (-56y + 8)$ Factor each group: $5x(7y - 1) - 8(7y - 1)$ Factor common term: $(5x - 8)(7y - 1)$

3. $x^2 + 15x + 50$

Find factors of $50$ that add to $15$: $10$ and $5$. $(x + 10)(x + 5)$

4. $c^2 + 2c - 24$

Find factors of $-24$ that add to $2$: $6$ and $-4$. $(c + 6)(c - 4)$

5. $9k^2 + 66k + 21$

Factor common terms first if applicable. Then find factors of $9 \times 21 = 189$ that add to $66$: $63$ and $3$. Split the middle term: $9k^2 + 63k + 3k + 21$ Group terms: $(9k^2 + 63k) + (3k + 21)$ Factor: $9k(k + 7) + 3(k + 7)$ Factor common term: $(9k + 3)(k + 7)$ Simplify: $3(3k + 1)(k + 7)$

6. $2n^2 + 5n + 2$

Find factors of $2 \times 2 = 4$ that add to $5$: $4$ and $1$. Split the middle term: $2n^2 + 4n + n + 2$ Group terms: $(2n^2 + 4n) + (n + 2)$ Factor: $2n(n + 2) + 1(n + 2)$ Factor common term: $(2n + 1)(n + 2)$

Summary of Factored Expressions:

1. $(5n^2 + 3)(n - 2)$
2. $(5x - 8)(7y - 1)$
3. $(x + 10)(x + 5)$
4. $(c + 6)(c - 4)$
5. $3(3k + 1)(k + 7)$
6. $(2n + 1)(n + 2)$

Would you like more details on any of these solutions or have other questions?

Related Questions:

1. How do you determine the best grouping strategy for factoring?
2. What are the common mistakes in factoring trinomials?
3. Can you explain why factoring is useful in solving equations?
4. What are the different methods for factoring polynomials?
5. How does factoring relate to graphing quadratic equations?

Tip:

Always check your factored expressions by expanding them to verify accuracy!