Factoring Trinomials (a is NOT 1) - Solve + Match Worksheet - Free Printable

Problem Overview:

The task involves factoring each given trinomial expression and then matching the factors with the provided options in the table. Each letter (A–I) corresponds to a factor, and we need to determine which letters match the factors of each trinomial.

Step-by-Step Solution:

#### 1. \( 3x^2 + 10x - 8 \)
Factor:
We need two numbers that multiply to \( 3 \cdot (-8) = -24 \) and add to \( 10 \). These numbers are \( 12 \) and \( -2 \).
Rewrite the middle term:
\[ 3x^2 + 12x - 2x - 8 \]
Group and factor by grouping:
\[ (3x^2 + 12x) + (-2x - 8) \]
\[ 3x(x + 4) - 2(x + 4) \]
Factor out the common binomial:
\[ (3x - 2)(x + 4) \]
Match:
The factors are \( (3x - 2) \) and \( (x + 4) \). From the table, \( (x + 4) \) corresponds to \( E \).

Answer for Problem 1: \( E \)

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#### 2. \( 4x^2 + 11x - 3 \)
Factor:
We need two numbers that multiply to \( 4 \cdot (-3) = -12 \) and add to \( 11 \). These numbers are \( 12 \) and \( -1 \).
Rewrite the middle term:
\[ 4x^2 + 12x - x - 3 \]
Group and factor by grouping:
\[ (4x^2 + 12x) + (-x - 3) \]
\[ 4x(x + 3) - 1(x + 3) \]
Factor out the common binomial:
\[ (4x - 1)(x + 3) \]
Match:
The factors are \( (4x - 1) \) and \( (x + 3) \). From the table, \( (x + 3) \) corresponds to \( D \).

Answer for Problem 2: \( D \)

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#### 3. \( 5x^2 + 19x + 18 \)
Factor:
We need two numbers that multiply to \( 5 \cdot 18 = 90 \) and add to \( 19 \). These numbers are \( 10 \) and \( 9 \).
Rewrite the middle term:
\[ 5x^2 + 10x + 9x + 18 \]
Group and factor by grouping:
\[ (5x^2 + 10x) + (9x + 18) \]
\[ 5x(x + 2) + 9(x + 2) \]
Factor out the common binomial:
\[ (5x + 9)(x + 2) \]
Match:
The factors are \( (5x + 9) \) and \( (x + 2) \). From the table, \( (x + 2) \) corresponds to \( H \).

Answer for Problem 3: \( H \)

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#### 4. \( 6x^2 - 29x - 5 \)
Factor:
We need two numbers that multiply to \( 6 \cdot (-5) = -30 \) and add to \( -29 \). These numbers are \( -30 \) and \( 1 \).
Rewrite the middle term:
\[ 6x^2 - 30x + x - 5 \]
Group and factor by grouping:
\[ (6x^2 - 30x) + (x - 5) \]
\[ 6x(x - 5) + 1(x - 5) \]
Factor out the common binomial:
\[ (6x + 1)(x - 5) \]
Match:
The factors are \( (6x + 1) \) and \( (x - 5) \). From the table, \( (x - 5) \) corresponds to \( I \).

Answer for Problem 4: \( I \)

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#### 5. \( 2x^2 - 17x + 21 \)
Factor:
We need two numbers that multiply to \( 2 \cdot 21 = 42 \) and add to \( -17 \). These numbers are \( -14 \) and \( -3 \).
Rewrite the middle term:
\[ 2x^2 - 14x - 3x + 21 \]
Group and factor by grouping:
\[ (2x^2 - 14x) + (-3x + 21) \]
\[ 2x(x - 7) - 3(x - 7) \]
Factor out the common binomial:
\[ (2x - 3)(x - 7) \]
Match:
The factors are \( (2x - 3) \) and \( (x - 7) \). From the table, \( (x - 7) \) corresponds to \( B \).

Answer for Problem 5: \( B \)

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#### 6. \( 7x^2 - 31x + 30 \)
Factor:
We need two numbers that multiply to \( 7 \cdot 30 = 210 \) and add to \( -31 \). These numbers are \( -21 \) and \( -10 \).
Rewrite the middle term:
\[ 7x^2 - 21x - 10x + 30 \]
Group and factor by grouping:
\[ (7x^2 - 21x) + (-10x + 30) \]
\[ 7x(x - 3) - 10(x - 3) \]
Factor out the common binomial:
\[ (7x - 10)(x - 3) \]
Match:
The factors are \( (7x - 10) \) and \( (x - 3) \). From the table, \( (x - 3) \) corresponds to \( G \).

Answer for Problem 6: \( G \)

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#### 7. \( 5x^2 + 11x + 6 \)
Factor:
We need two numbers that multiply to \( 5 \cdot 6 = 30 \) and add to \( 11 \). These numbers are \( 5 \) and \( 6 \).
Rewrite the middle term:
\[ 5x^2 + 5x + 6x + 6 \]
Group and factor by grouping:
\[ (5x^2 + 5x) + (6x + 6) \]
\[ 5x(x + 1) + 6(x + 1) \]
Factor out the common binomial:
\[ (5x + 6)(x + 1) \]
Match:
The factors are \( (5x + 6) \) and \( (x + 1) \). From the table, \( (x + 1) \) corresponds to \( A \).

Answer for Problem 7: \( A \)

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#### 8. \( 2x^2 - 7x - 30 \)
Factor:
We need two numbers that multiply to \( 2 \cdot (-30) = -60 \) and add to \( -7 \). These numbers are \( -15 \) and \( 8 \).
Rewrite the middle term:
\[ 2x^2 - 15x + 8x - 30 \]
Group and factor by grouping:
\[ (2x^2 - 15x) + (8x - 30) \]
\[ x(2x - 15) + 2(4x - 15) \]
Factor out the common binomial:
\[ (2x + 5)(x - 6) \]
Match:
The factors are \( (2x + 5) \) and \( (x - 6) \). From the table, \( (x - 6) \) corresponds to \( F \).

Answer for Problem 8: \( F \)

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#### 9. \( 4x^2 - x - 14 \)
Factor:
We need two numbers that multiply to \( 4 \cdot (-14) = -56 \) and add to \( -1 \). These numbers are \( -8 \) and \( 7 \).
Rewrite the middle term:
\[ 4x^2 - 8x + 7x - 14 \]
Group and factor by grouping:
\[ (4x^2 - 8x) + (7x - 14) \]
\[ 4x(x - 2) + 7(x - 2) \]
Factor out the common binomial:
\[ (4x + 7)(x - 2) \]
Match:
The factors are \( (4x + 7) \) and \( (x - 2) \). From the table, \( (x - 2) \) corresponds to \( C \).

Answer for Problem 9: \( C \)

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Final Answers:

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