Mathematics worksheet for Grade 8 students on factoring polynomials by grouping, provided by the Heisenberg Tutorial Center.

Here are the step-by-step solutions for each problem on the worksheet. The method used is factoring by grouping, which involves splitting the polynomial into two pairs, factoring out the greatest common factor (GCF) from each pair, and then factoring out the common binomial.

1) $4v^3 + 2v^2 - 10v - 5$
* Group terms: $(4v^3 + 2v^2)$ and $(-10v - 5)$.
* Factor GCF from first group: $2v^2(2v + 1)$.
* Factor GCF from second group: $-5(2v + 1)$.
* Combine: $(2v^2 - 5)(2v + 1)$.
* Matches Option B.

2) $21x^3 + 18x^2 - 14x - 12$
* Group terms: $(21x^3 + 18x^2)$ and $(-14x - 12)$.
* Factor GCF from first group: $3x^2(7x + 6)$.
* Factor GCF from second group: $-2(7x + 6)$.
* Combine: $(3x^2 - 2)(7x + 6)$.
* Matches Option C.

3) $15k^3 + 12k^2 - 10k - 8$
* Group terms: $(15k^3 + 12k^2)$ and $(-10k - 8)$.
* Factor GCF from first group: $3k^2(5k + 4)$.
* Factor GCF from second group: $-2(5k + 4)$.
* Combine: $(3k^2 - 2)(5k + 4)$.
* Matches Option D.

4) $3x^3 - 5x^2 + 9x - 15$
* Group terms: $(3x^3 - 5x^2)$ and $(9x - 15)$.
* Factor GCF from first group: $x^2(3x - 5)$.
* Factor GCF from second group: $3(3x - 5)$.
* Combine: $(x^2 + 3)(3x - 5)$.
* Matches Option B.

5) $21v^3 - 14v^2 - 9v + 6$
* Group terms: $(21v^3 - 14v^2)$ and $(-9v + 6)$.
* Factor GCF from first group: $7v^2(3v - 2)$.
* Factor GCF from second group: $-3(3v - 2)$.
* Combine: $(7v^2 - 3)(3v - 2)$.
* Matches Option D.

6) $12x^3 - 18x^2 - 2x + 3$
* Group terms: $(12x^3 - 18x^2)$ and $(-2x + 3)$.
* Factor GCF from first group: $6x^2(2x - 3)$.
* Factor GCF from second group: $-1(2x - 3)$.
* Combine: $(6x^2 - 1)(2x - 3)$.
* Matches Option A.

7) $10b^3 + 25b^2 + 12b + 30$
* Group terms: $(10b^3 + 25b^2)$ and $(12b + 30)$.
* Factor GCF from first group: $5b^2(2b + 5)$.
* Factor GCF from second group: $6(2b + 5)$.
* Combine: $(5b^2 + 6)(2b + 5)$.
* Matches Option B.

8) $35m^3 + 30m^2 - 7m - 6$
* Group terms: $(35m^3 + 30m^2)$ and $(-7m - 6)$.
* Factor GCF from first group: $5m^2(7m + 6)$.
* Factor GCF from second group: $-1(7m + 6)$.
* Combine: $(5m^2 - 1)(7m + 6)$.
* Matches Option C.

9) $28x^3 + 16x^2 - 49x - 28$
* Group terms: $(28x^3 + 16x^2)$ and $(-49x - 28)$.
* Factor GCF from first group: $4x^2(7x + 4)$.
* Factor GCF from second group: $-7(7x + 4)$.
* Combine: $(4x^2 - 7)(7x + 4)$.
* Matches Option D.

10) $4b^3 - 24b^2 + b - 6$
* Group terms: $(4b^3 - 24b^2)$ and $(b - 6)$.
* Factor GCF from first group: $4b^2(b - 6)$.
* Factor GCF from second group: $1(b - 6)$.
* Combine: $(4b^2 + 1)(b - 6)$.
* Matches Option C.

Final Answer:
1) B
2) C
3) D
4) B
5) D
6) A
7) B
8) C
9) D
10) C