Factoring trinomials practice worksheet with ten algebraic expressions to factor completely.

Let's solve each trinomial by factoring completely. We'll use the AC method (also known as the "splitting the middle term" method) for trinomials where the leading coefficient $ a > 1 $. The general form is:

$$
ax^2 + bx + c
$$

We look for two numbers that:
- Multiply to $ a \cdot c $
- Add to $ b $

Then we split the middle term and factor by grouping.

---

1. $ 4x^2 + 4x - 35 $

- $ a = 4, b = 4, c = -35 $
- $ a \cdot c = 4 \cdot (-35) = -140 $
- Find two numbers that multiply to $-140$ and add to $4$:
Try: $ 14 $ and $-10$: $ 14 \cdot (-10) = -140 $, $ 14 + (-10) = 4 $ ✔

Split the middle term:
$$
4x^2 + 14x - 10x - 35
$$

Group:
$$
(4x^2 + 14x) + (-10x - 35)
= 2x(2x + 7) -5(2x + 7)
= (2x - 5)(2x + 7)
$$

✔ Answer: $ (2x - 5)(2x + 7) $

---

2. $ 5x^2 + 37x - 24 $

- $ a = 5, b = 37, c = -24 $
- $ a \cdot c = 5 \cdot (-24) = -120 $
- Find two numbers that multiply to $-120$, add to $37$:
Try: $ 40 $ and $-3$: $ 40 \cdot (-3) = -120 $, $ 40 + (-3) = 37 $ ✔

Split:
$$
5x^2 + 40x - 3x - 24
$$

Group:
$$
(5x^2 + 40x) + (-3x - 24)
= 5x(x + 8) -3(x + 8)
= (5x - 3)(x + 8)
$$

✔ Answer: $ (5x - 3)(x + 8) $

---

3. $ 3x^2 + 12x + 12 $

First, factor out GCF: $ 3 $
$$
3(x^2 + 4x + 4)
$$

Now factor the trinomial inside:
$$
x^2 + 4x + 4 = (x + 2)^2
$$

✔ Answer: $ 3(x + 2)^2 $

---

4. $ 6x^2 - 44x + 14 $

Factor out GCF: $ 2 $
$$
2(3x^2 - 22x + 7)
$$

Now factor $ 3x^2 - 22x + 7 $:
- $ a = 3, b = -22, c = 7 $
- $ a \cdot c = 21 $
- Need two numbers that multiply to $21$, add to $-22$:
Try: $-21$ and $-1$: $-21 \cdot (-1) = 21$, $-21 + (-1) = -22$ ✔

Split:
$$
3x^2 - 21x - x + 7
$$

Group:
$$
(3x^2 - 21x) + (-x + 7)
= 3x(x - 7) -1(x - 7)
= (3x - 1)(x - 7)
$$

So overall:
$$
2(3x - 1)(x - 7)
$$

✔ Answer: $ 2(3x - 1)(x - 7) $

---

5. $ 3x^2 - 23x - 36 $

- $ a = 3, b = -23, c = -36 $
- $ a \cdot c = 3 \cdot (-36) = -108 $
- Need two numbers that multiply to $-108$, add to $-23$:
Try: $-27$ and $4$: $-27 \cdot 4 = -108$, $-27 + 4 = -23$ ✔

Split:
$$
3x^2 - 27x + 4x - 36
$$

Group:
$$
(3x^2 - 27x) + (4x - 36)
= 3x(x - 9) + 4(x - 9)
= (3x + 4)(x - 9)
$$

✔ Answer: $ (3x + 4)(x - 9) $

---

6. $ 2x^2 - 3x + 1 $

- $ a = 2, b = -3, c = 1 $
- $ a \cdot c = 2 $
- Need two numbers that multiply to $2$, add to $-3$:
Try: $-2$ and $-1$: $-2 \cdot (-1) = 2$, $-2 + (-1) = -3$ ✔

Split:
$$
2x^2 - 2x - x + 1
$$

Group:
$$
(2x^2 - 2x) + (-x + 1)
= 2x(x - 1) -1(x - 1)
= (2x - 1)(x - 1)
$$

✔ Answer: $ (2x - 1)(x - 1) $

---

7. $ 20x^2 + 2x - 6 $

Factor out GCF: $ 2 $
$$
2(10x^2 + x - 3)
$$

Now factor $ 10x^2 + x - 3 $:
- $ a = 10, b = 1, c = -3 $
- $ a \cdot c = -30 $
- Need two numbers that multiply to $-30$, add to $1$:
Try: $6$ and $-5$: $6 \cdot (-5) = -30$, $6 + (-5) = 1$ ✔

Split:
$$
10x^2 + 6x - 5x - 3
$$

Group:
$$
(10x^2 + 6x) + (-5x - 3)
= 2x(5x + 3) -1(5x + 3)
= (2x - 1)(5x + 3)
$$

So overall:
$$
2(2x - 1)(5x + 3)
$$

✔ Answer: $ 2(2x - 1)(5x + 3) $

---

8. $ 3x^2 + 25x + 42 $

- $ a = 3, b = 25, c = 42 $
- $ a \cdot c = 126 $
- Need two numbers that multiply to $126$, add to $25$:
Try: $18$ and $7$: $18 \cdot 7 = 126$, $18 + 7 = 25$ ✔

Split:
$$
3x^2 + 18x + 7x + 42
$$

Group:
$$
(3x^2 + 18x) + (7x + 42)
= 3x(x + 6) + 7(x + 6)
= (3x + 7)(x + 6)
$$

✔ Answer: $ (3x + 7)(x + 6) $

---

9. $ 2x^2 - 3x - 20 $

- $ a = 2, b = -3, c = -20 $
- $ a \cdot c = -40 $
- Need two numbers that multiply to $-40$, add to $-3$:
Try: $-8$ and $5$: $-8 \cdot 5 = -40$, $-8 + 5 = -3$ ✔

Split:
$$
2x^2 - 8x + 5x - 20
$$

Group:
$$
(2x^2 - 8x) + (5x - 20)
= 2x(x - 4) + 5(x - 4)
= (2x + 5)(x - 4)
$$

✔ Answer: $ (2x + 5)(x - 4) $

---

10. $ 35x^2 - 11x - 6 $

- $ a = 35, b = -11, c = -6 $
- $ a \cdot c = -210 $
- Need two numbers that multiply to $-210$, add to $-11$:
Try: $-21$ and $10$: $-21 \cdot 10 = -210$, $-21 + 10 = -11$ ✔

Split:
$$
35x^2 - 21x + 10x - 6
$$

Group:
$$
(35x^2 - 21x) + (10x - 6)
= 7x(5x - 3) + 2(5x - 3)
= (7x + 2)(5x - 3)
$$

✔ Answer: $ (7x + 2)(5x - 3) $

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✔ Final Answers:

| Problem | Factored Form |
|--------|----------------|
| 1. $ 4x^2 + 4x - 35 $ | $ (2x - 5)(2x + 7) $ |
| 2. $ 5x^2 + 37x - 24 $ | $ (5x - 3)(x + 8) $ |
| 3. $ 3x^2 + 12x + 12 $ | $ 3(x + 2)^2 $ |
| 4. $ 6x^2 - 44x + 14 $ | $ 2(3x - 1)(x - 7) $ |
| 5. $ 3x^2 - 23x - 36 $ | $ (3x + 4)(x - 9) $ |
| 6. $ 2x^2 - 3x + 1 $ | $ (2x - 1)(x - 1) $ |
| 7. $ 20x^2 + 2x - 6 $ | $ 2(2x - 1)(5x + 3) $ |
| 8. $ 3x^2 + 25x + 42 $ | $ (3x + 7)(x + 6) $ |
| 9. $ 2x^2 - 3x - 20 $ | $ (2x + 5)(x - 4) $ |
| 10. $ 35x^2 - 11x - 6 $ | $ (7x + 2)(5x - 3) $ |

Let me know if you'd like these checked with multiplication or explained further!