Worksheet for factoring quadratic expressions with positive coefficients and variables.

To factor each quadratic expression completely, we will use the method of factoring by grouping or by finding two numbers that multiply to give the product of the leading coefficient and the constant term, and add up to the middle coefficient. Let's solve each one step by step.

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1. Factor \( 5n^2 + 19n + 12 \)

We need to find two numbers that multiply to \( 5 \cdot 12 = 60 \) and add up to \( 19 \).

- The pairs of factors of 60 are: \( (1, 60), (2, 30), (3, 20), (4, 15), (5, 12), (6, 10) \).
- The pair \( (4, 15) \) adds up to 19.

Now, rewrite the middle term \( 19n \) as \( 4n + 15n \):

\[
5n^2 + 19n + 12 = 5n^2 + 4n + 15n + 12
\]

Group the terms:

\[
(5n^2 + 4n) + (15n + 12)
\]

Factor out the greatest common factor (GCF) from each group:

\[
n(5n + 4) + 3(5n + 4)
\]

Notice that \( (5n + 4) \) is a common factor:

\[
(n + 3)(5n + 4)
\]

So, the factored form is:

\[
\boxed{(n + 3)(5n + 4)}
\]

---

2. Factor \( 9k^2 + 66k + 21 \)

First, factor out the greatest common factor (GCF) of the coefficients, which is 3:

\[
9k^2 + 66k + 21 = 3(3k^2 + 22k + 7)
\]

Now, factor \( 3k^2 + 22k + 7 \). We need two numbers that multiply to \( 3 \cdot 7 = 21 \) and add up to 22.

- The pairs of factors of 21 are: \( (1, 21), (3, 7) \).
- The pair \( (1, 21) \) adds up to 22.

Rewrite the middle term \( 22k \) as \( k + 21k \):

\[
3k^2 + 22k + 7 = 3k^2 + k + 21k + 7
\]

Group the terms:

\[
(3k^2 + k) + (21k + 7)
\]

Factor out the GCF from each group:

\[
k(3k + 1) + 7(3k + 1)
\]

Notice that \( (3k + 1) \) is a common factor:

\[
(k + 7)(3k + 1)
\]

So, the factored form is:

\[
\boxed{3(k + 7)(3k + 1)}
\]

---

3. Factor \( 2n^2 + 5n + 2 \)

We need two numbers that multiply to \( 2 \cdot 2 = 4 \) and add up to 5.

- The pairs of factors of 4 are: \( (1, 4), (2, 2) \).
- The pair \( (1, 4) \) adds up to 5.

Rewrite the middle term \( 5n \) as \( n + 4n \):

\[
2n^2 + 5n + 2 = 2n^2 + n + 4n + 2
\]

Group the terms:

\[
(2n^2 + n) + (4n + 2)
\]

Factor out the GCF from each group:

\[
n(2n + 1) + 2(2n + 1)
\]

Notice that \( (2n + 1) \) is a common factor:

\[
(n + 2)(2n + 1)
\]

So, the factored form is:

\[
\boxed{(n + 2)(2n + 1)}
\]

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4. Factor \( 6x^2 + 37x + 6 \)

We need two numbers that multiply to \( 6 \cdot 6 = 36 \) and add up to 37.

- The pairs of factors of 36 are: \( (1, 36), (2, 18), (3, 12), (4, 9), (6, 6) \).
- The pair \( (1, 36) \) adds up to 37.

Rewrite the middle term \( 37x \) as \( x + 36x \):

\[
6x^2 + 37x + 6 = 6x^2 + x + 36x + 6
\]

Group the terms:

\[
(6x^2 + x) + (36x + 6)
\]

Factor out the GCF from each group:

\[
x(6x + 1) + 6(6x + 1)
\]

Notice that \( (6x + 1) \) is a common factor:

\[
(x + 6)(6x + 1)
\]

So, the factored form is:

\[
\boxed{(x + 6)(6x + 1)}
\]

---

5. Factor \( 2v^2 + 11v + 5 \)

We need two numbers that multiply to \( 2 \cdot 5 = 10 \) and add up to 11.

- The pairs of factors of 10 are: \( (1, 10), (2, 5) \).
- The pair \( (1, 10) \) adds up to 11.

Rewrite the middle term \( 11v \) as \( v + 10v \):

\[
2v^2 + 11v + 5 = 2v^2 + v + 10v + 5
\]

Group the terms:

\[
(2v^2 + v) + (10v + 5)
\]

Factor out the GCF from each group:

\[
v(2v + 1) + 5(2v + 1)
\]

Notice that \( (2v + 1) \) is a common factor:

\[
(v + 5)(2v + 1)
\]

So, the factored form is:

\[
\boxed{(v + 5)(2v + 1)}
\]

---

6. Factor \( 2b^2 + 17b + 21 \)

We need two numbers that multiply to \( 2 \cdot 21 = 42 \) and add up to 17.

- The pairs of factors of 42 are: \( (1, 42), (2, 21), (3, 14), (6, 7) \).
- The pair \( (6, 7) \) adds up to 17.

Rewrite the middle term \( 17b \) as \( 6b + 7b \):

\[
2b^2 + 17b + 21 = 2b^2 + 6b + 7b + 21
\]

Group the terms:

\[
(2b^2 + 6b) + (7b + 21)
\]

Factor out the GCF from each group:

\[
2b(b + 3) + 7(b + 3)
\]

Notice that \( (b + 3) \) is a common factor:

\[
(2b + 7)(b + 3)
\]

So, the factored form is:

\[
\boxed{(2b + 7)(b + 3)}
\]

---

7. Factor \( 7a^2 + 53a + 28 \)

We need two numbers that multiply to \( 7 \cdot 28 = 196 \) and add up to 53.

- The pairs of factors of 196 are: \( (1, 196), (2, 98), (4, 49), (7, 28), (14, 14) \).
- The pair \( (4, 49) \) adds up to 53.

Rewrite the middle term \( 53a \) as \( 4a + 49a \):

\[
7a^2 + 53a + 28 = 7a^2 + 4a + 49a + 28
\]

Group the terms:

\[
(7a^2 + 4a) + (49a + 28)
\]

Factor out the GCF from each group:

\[
a(7a + 4) + 7(7a + 4)
\]

Notice that \( (7a + 4) \) is a common factor:

\[
(a + 7)(7a + 4)
\]

So, the factored form is:

\[
\boxed{(a + 7)(7a + 4)}
\]

---

8. Factor \( 9p^2 + 73p + 70 \)

We need two numbers that multiply to \( 9 \cdot 70 = 630 \) and add up to 73.

- The pairs of factors of 630 are: \( (1, 630), (2, 315), (3, 210), (5, 126), (6, 105), (7, 90), (9, 70), (10, 63), (14, 45), (15, 42), (18, 35), (21, 30) \).
- The pair \( (7, 90) \) adds up to 73.

Rewrite the middle term \( 73p \) as \( 7p + 90p \):

\[
9p^2 + 73p + 70 = 9p^2 + 7p + 90p + 70
\]

Group the terms:

\[
(9p^2 + 7p) + (90p + 70)
\]

Factor out the GCF from each group:

\[
p(9p + 7) + 10(9p + 7)
\]

Notice that \( (9p + 7) \) is a common factor:

\[
(p + 10)(9p + 7)
\]

So, the factored form is:

\[
\boxed{(p + 10)(9p + 7)}
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ (n + 3)(5n + 4) \\
2. & \ 3(k + 7)(3k + 1) \\
3. & \ (n + 2)(2n + 1) \\
4. & \ (x + 6)(6x + 1) \\
5. & \ (v + 5)(2v + 1) \\
6. & \ (2b + 7)(b + 3) \\
7. & \ (a + 7)(7a + 4) \\
8. & \ (p + 10)(9p + 7)
\end{aligned}
}
\]