Factoring trinomials practice worksheet for algebra students.

To solve the problem of factoring each trinomial completely, we will use various techniques such as finding factors of the leading coefficient and constant term, grouping, and identifying patterns. Let's go through each expression step by step.

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1. \( 4x^2 + 4x - 35 \)

We need to factor \( 4x^2 + 4x - 35 \). The general form is \( ax^2 + bx + c \), where \( a = 4 \), \( b = 4 \), and \( c = -35 \).

#### Step 1: Find two numbers that multiply to \( ac \) and add to \( b \).
- \( ac = 4 \cdot (-35) = -140 \)
- We need two numbers that multiply to \( -140 \) and add to \( 4 \).

The pair of numbers is \( 14 \) and \( -10 \) because:
\[ 14 \cdot (-10) = -140 \]
\[ 14 + (-10) = 4 \]

#### Step 2: Rewrite the middle term using these numbers.
\[ 4x^2 + 4x - 35 = 4x^2 + 14x - 10x - 35 \]

#### Step 3: Group the terms and factor by grouping.
\[ (4x^2 + 14x) + (-10x - 35) \]
\[ 2x(2x + 7) - 5(2x + 7) \]

#### Step 4: Factor out the common binomial factor.
\[ (2x + 7)(2x - 5) \]

So, the factored form is:
\[ \boxed{(2x + 7)(2x - 5)} \]

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2. \( 5x^2 + 37x - 24 \)

We need to factor \( 5x^2 + 37x - 24 \). The general form is \( ax^2 + bx + c \), where \( a = 5 \), \( b = 37 \), and \( c = -24 \).

#### Step 1: Find two numbers that multiply to \( ac \) and add to \( b \).
- \( ac = 5 \cdot (-24) = -120 \)
- We need two numbers that multiply to \( -120 \) and add to \( 37 \).

The pair of numbers is \( 40 \) and \( -3 \) because:
\[ 40 \cdot (-3) = -120 \]
\[ 40 + (-3) = 37 \]

#### Step 2: Rewrite the middle term using these numbers.
\[ 5x^2 + 37x - 24 = 5x^2 + 40x - 3x - 24 \]

#### Step 3: Group the terms and factor by grouping.
\[ (5x^2 + 40x) + (-3x - 24) \]
\[ 5x(x + 8) - 3(x + 8) \]

#### Step 4: Factor out the common binomial factor.
\[ (5x - 3)(x + 8) \]

So, the factored form is:
\[ \boxed{(5x - 3)(x + 8)} \]

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3. \( 3x^2 + 12x + 12 \)

We need to factor \( 3x^2 + 12x + 12 \). First, notice that all terms have a common factor of 3.

#### Step 1: Factor out the greatest common factor (GCF).
\[ 3x^2 + 12x + 12 = 3(x^2 + 4x + 4) \]

#### Step 2: Factor the quadratic expression inside the parentheses.
\[ x^2 + 4x + 4 = (x + 2)^2 \]

So, the factored form is:
\[ \boxed{3(x + 2)^2} \]

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4. \( 6x^2 - 44x + 14 \)

We need to factor \( 6x^2 - 44x + 14 \). First, notice that all terms have a common factor of 2.

#### Step 1: Factor out the greatest common factor (GCF).
\[ 6x^2 - 44x + 14 = 2(3x^2 - 22x + 7) \]

#### Step 2: Factor the quadratic expression inside the parentheses.
We need to factor \( 3x^2 - 22x + 7 \). The general form is \( ax^2 + bx + c \), where \( a = 3 \), \( b = -22 \), and \( c = 7 \).

- \( ac = 3 \cdot 7 = 21 \)
- We need two numbers that multiply to \( 21 \) and add to \( -22 \).

The pair of numbers is \( -21 \) and \( -1 \) because:
\[ -21 \cdot (-1) = 21 \]
\[ -21 + (-1) = -22 \]

#### Step 3: Rewrite the middle term using these numbers.
\[ 3x^2 - 22x + 7 = 3x^2 - 21x - x + 7 \]

#### Step 4: Group the terms and factor by grouping.
\[ (3x^2 - 21x) + (-x + 7) \]
\[ 3x(x - 7) - 1(x - 7) \]

#### Step 5: Factor out the common binomial factor.
\[ (3x - 1)(x - 7) \]

So, the factored form is:
\[ \boxed{2(3x - 1)(x - 7)} \]

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5. \( 3x^2 - 23x - 36 \)

We need to factor \( 3x^2 - 23x - 36 \). The general form is \( ax^2 + bx + c \), where \( a = 3 \), \( b = -23 \), and \( c = -36 \).

#### Step 1: Find two numbers that multiply to \( ac \) and add to \( b \).
- \( ac = 3 \cdot (-36) = -108 \)
- We need two numbers that multiply to \( -108 \) and add to \( -23 \).

The pair of numbers is \( -27 \) and \( 4 \) because:
\[ -27 \cdot 4 = -108 \]
\[ -27 + 4 = -23 \]

#### Step 2: Rewrite the middle term using these numbers.
\[ 3x^2 - 23x - 36 = 3x^2 - 27x + 4x - 36 \]

#### Step 3: Group the terms and factor by grouping.
\[ (3x^2 - 27x) + (4x - 36) \]
\[ 3x(x - 9) + 4(x - 9) \]

#### Step 4: Factor out the common binomial factor.
\[ (3x + 4)(x - 9) \]

So, the factored form is:
\[ \boxed{(3x + 4)(x - 9)} \]

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6. \( 2x^2 - 3x + 1 \)

We need to factor \( 2x^2 - 3x + 1 \). The general form is \( ax^2 + bx + c \), where \( a = 2 \), \( b = -3 \), and \( c = 1 \).

#### Step 1: Find two numbers that multiply to \( ac \) and add to \( b \).
- \( ac = 2 \cdot 1 = 2 \)
- We need two numbers that multiply to \( 2 \) and add to \( -3 \).

The pair of numbers is \( -2 \) and \( -1 \) because:
\[ -2 \cdot (-1) = 2 \]
\[ -2 + (-1) = -3 \]

#### Step 2: Rewrite the middle term using these numbers.
\[ 2x^2 - 3x + 1 = 2x^2 - 2x - x + 1 \]

#### Step 3: Group the terms and factor by grouping.
\[ (2x^2 - 2x) + (-x + 1) \]
\[ 2x(x - 1) - 1(x - 1) \]

#### Step 4: Factor out the common binomial factor.
\[ (2x - 1)(x - 1) \]

So, the factored form is:
\[ \boxed{(2x - 1)(x - 1)} \]

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7. \( 20x^2 + 2x - 6 \)

We need to factor \( 20x^2 + 2x - 6 \). First, notice that all terms have a common factor of 2.

#### Step 1: Factor out the greatest common factor (GCF).
\[ 20x^2 + 2x - 6 = 2(10x^2 + x - 3) \]

#### Step 2: Factor the quadratic expression inside the parentheses.
We need to factor \( 10x^2 + x - 3 \). The general form is \( ax^2 + bx + c \), where \( a = 10 \), \( b = 1 \), and \( c = -3 \).

- \( ac = 10 \cdot (-3) = -30 \)
- We need two numbers that multiply to \( -30 \) and add to \( 1 \).

The pair of numbers is \( 6 \) and \( -5 \) because:
\[ 6 \cdot (-5) = -30 \]
\[ 6 + (-5) = 1 \]

#### Step 3: Rewrite the middle term using these numbers.
\[ 10x^2 + x - 3 = 10x^2 + 6x - 5x - 3 \]

#### Step 4: Group the terms and factor by grouping.
\[ (10x^2 + 6x) + (-5x - 3) \]
\[ 2x(5x + 3) - 1(5x + 3) \]

#### Step 5: Factor out the common binomial factor.
\[ (2x - 1)(5x + 3) \]

So, the factored form is:
\[ \boxed{2(2x - 1)(5x + 3)} \]

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8. \( 3x^2 + 25x + 42 \)

We need to factor \( 3x^2 + 25x + 42 \). The general form is \( ax^2 + bx + c \), where \( a = 3 \), \( b = 25 \), and \( c = 42 \).

#### Step 1: Find two numbers that multiply to \( ac \) and add to \( b \).
- \( ac = 3 \cdot 42 = 126 \)
- We need two numbers that multiply to \( 126 \) and add to \( 25 \).

The pair of numbers is \( 18 \) and \( 7 \) because:
\[ 18 \cdot 7 = 126 \]
\[ 18 + 7 = 25 \]

#### Step 2: Rewrite the middle term using these numbers.
\[ 3x^2 + 25x + 42 = 3x^2 + 18x + 7x + 42 \]

#### Step 3: Group the terms and factor by grouping.
\[ (3x^2 + 18x) + (7x + 42) \]
\[ 3x(x + 6) + 7(x + 6) \]

#### Step 4: Factor out the common binomial factor.
\[ (3x + 7)(x + 6) \]

So, the factored form is:
\[ \boxed{(3x + 7)(x + 6)} \]

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9. \( 2x^2 - 3x - 20 \)

We need to factor \( 2x^2 - 3x - 20 \). The general form is \( ax^2 + bx + c \), where \( a = 2 \), \( b = -3 \), and \( c = -20 \).

#### Step 1: Find two numbers that multiply to \( ac \) and add to \( b \).
- \( ac = 2 \cdot (-20) = -40 \)
- We need two numbers that multiply to \( -40 \) and add to \( -3 \).

The pair of numbers is \( -8 \) and \( 5 \) because:
\[ -8 \cdot 5 = -40 \]
\[ -8 + 5 = -3 \]

#### Step 2: Rewrite the middle term using these numbers.
\[ 2x^2 - 3x - 20 = 2x^2 - 8x + 5x - 20 \]

#### Step 3: Group the terms and factor by grouping.
\[ (2x^2 - 8x) + (5x - 20) \]
\[ 2x(x - 4) + 5(x - 4) \]

#### Step 4: Factor out the common binomial factor.
\[ (2x + 5)(x - 4) \]

So, the factored form is:
\[ \boxed{(2x + 5)(x - 4)} \]

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10. \( 35x^2 - 11x - 6 \)

We need to factor \( 35x^2 - 11x - 6 \). The general form is \( ax^2 + bx + c \), where \( a = 35 \), \( b = -11 \), and \( c = -6 \).

#### Step 1: Find two numbers that multiply to \( ac \) and add to \( b \).
- \( ac = 35 \cdot (-6) = -210 \)
- We need two numbers that multiply to \( -210 \) and add to \( -11 \).

The pair of numbers is \( -21 \) and \( 10 \) because:
\[ -21 \cdot 10 = -210 \]
\[ -21 + 10 = -11 \]

#### Step 2: Rewrite the middle term using these numbers.
\[ 35x^2 - 11x - 6 = 35x^2 - 21x + 10x - 6 \]

#### Step 3: Group the terms and factor by grouping.
\[ (35x^2 - 21x) + (10x - 6) \]
\[ 7x(5x - 3) + 2(5x - 3) \]

#### Step 4: Factor out the common binomial factor.
\[ (7x + 2)(5x - 3) \]

So, the factored form is:
\[ \boxed{(7x + 2)(5x - 3)} \]

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Final Answers:

1. \( \boxed{(2x + 7)(2x - 5)} \)
2. \( \boxed{(5x - 3)(x + 8)} \)
3. \( \boxed{3(x + 2)^2} \)
4. \( \boxed{2(3x - 1)(x - 7)} \)
5. \( \boxed{(3x + 4)(x - 9)} \)
6. \( \boxed{(2x - 1)(x - 1)} \)
7. \( \boxed{2(2x - 1)(5x + 3)} \)
8. \( \boxed{(3x + 7)(x + 6)} \)
9. \( \boxed{(2x + 5)(x - 4)} \)
10. \( \boxed{(7x + 2)(5x - 3)} \)