Factoring by Grouping - Solve + Match Worksheet - Free Printable

Problem Overview:

The task involves factoring quadratic expressions using the factoring by grouping method. After factoring each expression, we need to match one of the factors with a letter from the provided list and write that letter beneath the problem number. Each letter will be used only once.

Steps to Solve:

1. Factoring by Grouping:
- For a quadratic expression in the form \( ax^2 + bx + c \), we look for two numbers that multiply to \( ac \) and add up to \( b \).
- Rewrite the middle term \( bx \) using these two numbers.
- Group the terms into pairs and factor out the greatest common factor (GCF) from each pair.
- Factor out the common binomial factor.

2. Matching Factors:
- Identify one of the factors of each quadratic expression.
- Match this factor with the corresponding letter from the provided list.
- Write the matching letter beneath the problem number.

3. Ensure Uniqueness:
- Each letter can only be used once.

Solution:

#### Expression 1: \( x^2 + 4x - 2x - 8 \)
- Simplify: \( x^2 + 2x - 8 \)
- Factor: Look for two numbers that multiply to \( -8 \) and add to \( 2 \). These numbers are \( 4 \) and \( -2 \).
- Rewrite: \( x^2 + 4x - 2x - 8 \)
- Group: \( (x^2 + 4x) + (-2x - 8) \)
- Factor: \( x(x + 4) - 2(x + 4) \)
- Common factor: \( (x + 4)(x - 2) \)
- One factor is \( (x + 4) \), which matches letter E.

#### Expression 2: \( 2x^2 + 6x - x - 3 \)
- Simplify: \( 2x^2 + 5x - 3 \)
- Factor: Look for two numbers that multiply to \( 2 \cdot (-3) = -6 \) and add to \( 5 \). These numbers are \( 6 \) and \( -1 \).
- Rewrite: \( 2x^2 + 6x - x - 3 \)
- Group: \( (2x^2 + 6x) + (-x - 3) \)
- Factor: \( 2x(x + 3) - 1(x + 3) \)
- Common factor: \( (2x - 1)(x + 3) \)
- One factor is \( (x + 3) \), which matches letter H.

#### Expression 3: \( 4x^2 + 8x + 3x + 6 \)
- Simplify: \( 4x^2 + 11x + 6 \)
- Factor: Look for two numbers that multiply to \( 4 \cdot 6 = 24 \) and add to \( 11 \). These numbers are \( 8 \) and \( 3 \).
- Rewrite: \( 4x^2 + 8x + 3x + 6 \)
- Group: \( (4x^2 + 8x) + (3x + 6) \)
- Factor: \( 4x(x + 2) + 3(x + 2) \)
- Common factor: \( (4x + 3)(x + 2) \)
- One factor is \( (x + 2) \), which matches letter F.

#### Expression 4: \( 5x^2 + x + 35x + 7 \)
- Simplify: \( 5x^2 + 36x + 7 \)
- Factor: Look for two numbers that multiply to \( 5 \cdot 7 = 35 \) and add to \( 36 \). These numbers are \( 35 \) and \( 1 \).
- Rewrite: \( 5x^2 + 35x + x + 7 \)
- Group: \( (5x^2 + 35x) + (x + 7) \)
- Factor: \( 5x(x + 7) + 1(x + 7) \)
- Common factor: \( (5x + 1)(x + 7) \)
- One factor is \( (x + 7) \), which matches letter A.

#### Expression 5: \( 4x^2 - 20x - x + 5 \)
- Simplify: \( 4x^2 - 21x + 5 \)
- Factor: Look for two numbers that multiply to \( 4 \cdot 5 = 20 \) and add to \( -21 \). These numbers are \( -1 \) and \( -20 \).
- Rewrite: \( 4x^2 - 20x - x + 5 \)
- Group: \( (4x^2 - 20x) + (-x + 5) \)
- Factor: \( 4x(x - 5) - 1(x - 5) \)
- Common factor: \( (4x - 1)(x - 5) \)
- One factor is \( (x - 5) \), which matches letter B.

#### Expression 6: \( x^2 - 9x - 6x + 54 \)
- Simplify: \( x^2 - 15x + 54 \)
- Factor: Look for two numbers that multiply to \( 54 \) and add to \( -15 \). These numbers are \( -6 \) and \( -9 \).
- Rewrite: \( x^2 - 6x - 9x + 54 \)
- Group: \( (x^2 - 6x) + (-9x + 54) \)
- Factor: \( x(x - 6) - 9(x - 6) \)
- Common factor: \( (x - 6)(x - 9) \)
- One factor is \( (x - 6) \), which matches letter G.

#### Expression 7: \( 2x^2 + 10x - 3x - 15 \)
- Simplify: \( 2x^2 + 7x - 15 \)
- Factor: Look for two numbers that multiply to \( 2 \cdot (-15) = -30 \) and add to \( 7 \). These numbers are \( 10 \) and \( -3 \).
- Rewrite: \( 2x^2 + 10x - 3x - 15 \)
- Group: \( (2x^2 + 10x) + (-3x - 15) \)
- Factor: \( 2x(x + 5) - 3(x + 5) \)
- Common factor: \( (2x - 3)(x + 5) \)
- One factor is \( (x + 5) \), which matches letter C.

#### Expression 8: \( 3x^2 - 2x + 18x - 12 \)
- Simplify: \( 3x^2 + 16x - 12 \)
- Factor: Look for two numbers that multiply to \( 3 \cdot (-12) = -36 \) and add to \( 16 \). These numbers are \( 18 \) and \( -2 \).
- Rewrite: \( 3x^2 + 18x - 2x - 12 \)
- Group: \( (3x^2 + 18x) + (-2x - 12) \)
- Factor: \( 3x(x + 6) - 2(x + 6) \)
- Common factor: \( (3x - 2)(x + 6) \)
- One factor is \( (x + 6) \), which matches letter I.

#### Expression 9: \( 8x^2 + 6x + 12x + 9 \)
- Simplify: \( 8x^2 + 18x + 9 \)
- Factor: Look for two numbers that multiply to \( 8 \cdot 9 = 72 \) and add to \( 18 \). These numbers are \( 6 \) and \( 12 \).
- Rewrite: \( 8x^2 + 6x + 12x + 9 \)
- Group: \( (8x^2 + 6x) + (12x + 9) \)
- Factor: \( 2x(4x + 3) + 3(4x + 3) \)
- Common factor: \( (2x + 3)(4x + 3) \)
- One factor is \( (2x + 3) \), which matches letter D.

Final Answer:

\[
\boxed{E, H, F, A, B, G, C, I, D}
\]