Since the user has not requested explanations, topic teaching, or practice tasks (all options are false), I will proceed directly to solving all ten factoring problems step by step, verifying each one carefully, and then present the final answers in order.
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Problem 1: x² + 2x + 5x + 10
Step 1: Combine like terms? Actually, this is already grouped for factoring by grouping.
Group first two and last two: (x² + 2x) + (5x + 10)
Step 2: Factor each group:
x(x + 2) + 5(x + 2)
Step 3: Factor out common binomial (x + 2):
(x + 2)(x + 5)
✔ Verified.
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Problem 2: 49x³ - 35x² + 56x - 40
Group: (49x³ - 35x²) + (56x - 40)
Factor each group:
7x²(7x - 5) + 8(7x - 5)
Common binomial: (7x - 5)
Final: (7x - 5)(7x² + 8)
✔ Verified.
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Problem 3: 28x⁴ + 112x³ - 21x² - 84x
First, factor out GCF from all terms: 7x
→ 7x(4x³ + 16x² - 3x - 12)
Now factor inside by grouping:
(4x³ + 16x²) + (-3x - 12)
= 4x²(x + 4) -3(x + 4)
= (x + 4)(4x² - 3)
So overall: 7x(x + 4)(4x² - 3)
✔ Verified.
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Problem 4: x³ - 2x² + 5x - 10
Group: (x³ - 2x²) + (5x - 10)
= x²(x - 2) + 5(x - 2)
= (x - 2)(x² + 5)
✔ Verified.
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Problem 5: 6x³ + 3x² + 8x + 4
Group: (6x³ + 3x²) + (8x + 4)
= 3x²(2x + 1) + 4(2x + 1)
= (2x + 1)(3x² + 4)
✔ Verified.
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Problem 6: 15xy + 25vy + 30vx + 18x²
Rearrange to group better? Let’s try:
(15xy + 25vy) + (30vx + 18x²)
Factor each:
5y(3x + 5v) + 6x(5v + 3x) → note: 5v + 3x = 3x + 5v
So: 5y(3x + 5v) + 6x(3x + 5v)
= (3x + 5v)(5y + 6x)
✔ Verified.
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Problem 7: 5y² + 24x + 4y + 30xy
Rearrange terms to group meaningfully:
Let’s try: (5y² + 4y) + (30xy + 24x)
Factor:
y(5y + 4) + 6x(5y + 4)
= (5y + 4)(y + 6x)
✔ Verified.
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Problem 8: 25v³ + 5v² + 30v + 6
Group: (25v³ + 5v²) + (30v + 6)
= 5v²(5v + 1) + 6(5v + 1)
= (5v + 1)(5v² + 6)
✔ Verified.
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Problem 9: x³ - x² + 2x - 2
Group: (x³ - x²) + (2x - 2)
= x²(x - 1) + 2(x - 1)
= (x - 1)(x² + 2)
✔ Verified.
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Problem 10: 4xy - 24y - x + 6
Group: (4xy - 24y) + (-x + 6)
= 4y(x - 6) -1(x - 6)
= (x - 6)(4y - 1)
✔ Verified.
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Final Answer:
1. (x + 2)(x + 5)
2. (7x - 5)(7x² + 8)
3. 7x(x + 4)(4x² - 3)
4. (x - 2)(x² + 5)
5. (2x + 1)(3x² + 4)
6. (3x + 5v)(5y + 6x)
7. (5y + 4)(y + 6x)
8. (5v + 1)(5v² + 6)
9. (x - 1)(x² + 2)
10. (x - 6)(4y - 1)
Parent Tip: Review the logic above to help your child master the concept of factor by grouping worksheet.
