Factoring General Trinomials - Free Printable

Problem:

The task is to factor all the given polynomials completely. Below, I will solve each polynomial step by step using appropriate factoring techniques such as:

1. Factoring out the Greatest Common Factor (GCF),
2. Factoring trinomials,
3. Difference of squares,
4. Perfect square trinomials,
5. Grouping, and
6. Trial and error for quadratic expressions.

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Solution:

#### 1. \( 5x^2 + 8x + 3 \)
- This is a trinomial of the form \( ax^2 + bx + c \).
- We need two numbers that multiply to \( 5 \cdot 3 = 15 \) and add to \( 8 \).
- The numbers are \( 5 \) and \( 3 \).
- Rewrite the middle term: \( 5x^2 + 5x + 3x + 3 \).
- Group and factor: \( (5x^2 + 5x) + (3x + 3) = 5x(x + 1) + 3(x + 1) \).
- Factor out the common binomial: \( (5x + 3)(x + 1) \).

Answer: \( (5x + 3)(x + 1) \)

#### 2. \( 5x^2 - 2x - 3 \)
- Again, this is a trinomial.
- We need two numbers that multiply to \( 5 \cdot (-3) = -15 \) and add to \( -2 \).
- The numbers are \( -5 \) and \( 3 \).
- Rewrite the middle term: \( 5x^2 - 5x + 3x - 3 \).
- Group and factor: \( (5x^2 - 5x) + (3x - 3) = 5x(x - 1) + 3(x - 1) \).
- Factor out the common binomial: \( (5x + 3)(x - 1) \).

Answer: \( (5x + 3)(x - 1) \)

#### 3. \( 5x^2 - 8x + 3 \)
- This is another trinomial.
- We need two numbers that multiply to \( 5 \cdot 3 = 15 \) and add to \( -8 \).
- The numbers are \( -5 \) and \( -3 \).
- Rewrite the middle term: \( 5x^2 - 5x - 3x + 3 \).
- Group and factor: \( (5x^2 - 5x) - (3x - 3) = 5x(x - 1) - 3(x - 1) \).
- Factor out the common binomial: \( (5x - 3)(x - 1) \).

Answer: \( (5x - 3)(x - 1) \)

#### 4. \( 5x^2 + 2x - 3 \)
- This is a trinomial.
- We need two numbers that multiply to \( 5 \cdot (-3) = -15 \) and add to \( 2 \).
- The numbers are \( 5 \) and \( -3 \).
- Rewrite the middle term: \( 5x^2 + 5x - 3x - 3 \).
- Group and factor: \( (5x^2 + 5x) - (3x + 3) = 5x(x + 1) - 3(x + 1) \).
- Factor out the common binomial: \( (5x - 3)(x + 1) \).

Answer: \( (5x - 3)(x + 1) \)

#### 5. \( x^2 + 6x - 7 \)
- This is a simpler trinomial.
- We need two numbers that multiply to \( 1 \cdot (-7) = -7 \) and add to \( 6 \).
- The numbers are \( 7 \) and \( -1 \).
- Rewrite the middle term: \( x^2 + 7x - x - 7 \).
- Group and factor: \( (x^2 + 7x) - (x + 7) = x(x + 7) - 1(x + 7) \).
- Factor out the common binomial: \( (x + 7)(x - 1) \).

Answer: \( (x + 7)(x - 1) \)

#### 6. \( x^2 - 6x - 7 \)
- This is a trinomial.
- We need two numbers that multiply to \( 1 \cdot (-7) = -7 \) and add to \( -6 \).
- The numbers are \( -7 \) and \( 1 \).
- Rewrite the middle term: \( x^2 - 7x + x - 7 \).
- Group and factor: \( (x^2 - 7x) + (x - 7) = x(x - 7) + 1(x - 7) \).
- Factor out the common binomial: \( (x - 7)(x + 1) \).

Answer: \( (x - 7)(x + 1) \)

#### 7. \( x^2 - 8x + 7 \)
- This is a trinomial.
- We need two numbers that multiply to \( 1 \cdot 7 = 7 \) and add to \( -8 \).
- The numbers are \( -7 \) and \( -1 \).
- Rewrite the middle term: \( x^2 - 7x - x + 7 \).
- Group and factor: \( (x^2 - 7x) - (x - 7) = x(x - 7) - 1(x - 7) \).
- Factor out the common binomial: \( (x - 7)(x - 1) \).

Answer: \( (x - 7)(x - 1) \)

#### 8. \( 7x^2 + 10x + 3 \)
- This is a trinomial.
- We need two numbers that multiply to \( 7 \cdot 3 = 21 \) and add to \( 10 \).
- The numbers are \( 7 \) and \( 3 \).
- Rewrite the middle term: \( 7x^2 + 7x + 3x + 3 \).
- Group and factor: \( (7x^2 + 7x) + (3x + 3) = 7x(x + 1) + 3(x + 1) \).
- Factor out the common binomial: \( (7x + 3)(x + 1) \).

Answer: \( (7x + 3)(x + 1) \)

#### 9. \( 7x^2 - 10x + 3 \)
- This is a trinomial.
- We need two numbers that multiply to \( 7 \cdot 3 = 21 \) and add to \( -10 \).
- The numbers are \( -7 \) and \( -3 \).
- Rewrite the middle term: \( 7x^2 - 7x - 3x + 3 \).
- Group and factor: \( (7x^2 - 7x) - (3x - 3) = 7x(x - 1) - 3(x - 1) \).
- Factor out the common binomial: \( (7x - 3)(x - 1) \).

Answer: \( (7x - 3)(x - 1) \)

#### 10. \( 7x^2 + 4x - 3 \)
- This is a trinomial.
- We need two numbers that multiply to \( 7 \cdot (-3) = -21 \) and add to \( 4 \).
- The numbers are \( 7 \) and \( -3 \).
- Rewrite the middle term: \( 7x^2 + 7x - 3x - 3 \).
- Group and factor: \( (7x^2 + 7x) - (3x + 3) = 7x(x + 1) - 3(x + 1) \).
- Factor out the common binomial: \( (7x - 3)(x + 1) \).

Answer: \( (7x - 3)(x + 1) \)

#### 11. \( 7x^2 + 22x + 3 \)
- This is a trinomial.
- We need two numbers that multiply to \( 7 \cdot 3 = 21 \) and add to \( 22 \).
- The numbers are \( 21 \) and \( 1 \).
- Rewrite the middle term: \( 7x^2 + 21x + x + 3 \).
- Group and factor: \( (7x^2 + 21x) + (x + 3) = 7x(x + 3) + 1(x + 3) \).
- Factor out the common binomial: \( (7x + 1)(x + 3) \).

Answer: \( (7x + 1)(x + 3) \)

#### 12. \( 7x^2 + 20x - 3 \)
- This is a trinomial.
- We need two numbers that multiply to \( 7 \cdot (-3) = -21 \) and add to \( 20 \).
- The numbers are \( 21 \) and \( -1 \).
- Rewrite the middle term: \( 7x^2 + 21x - x - 3 \).
- Group and factor: \( (7x^2 + 21x) - (x + 3) = 7x(x + 3) - 1(x + 3) \).
- Factor out the common binomial: \( (7x - 1)(x + 3) \).

Answer: \( (7x - 1)(x + 3) \)

#### 13. \( 7x^2 - 22x + 3 \)
- This is a trinomial.
- We need two numbers that multiply to \( 7 \cdot 3 = 21 \) and add to \( -22 \).
- The numbers are \( -21 \) and \( -1 \).
- Rewrite the middle term: \( 7x^2 - 21x - x + 3 \).
- Group and factor: \( (7x^2 - 21x) - (x - 3) = 7x(x - 3) - 1(x - 3) \).
- Factor out the common binomial: \( (7x - 1)(x - 3) \).

Answer: \( (7x - 1)(x - 3) \)

#### 14. \( 9x^2 - 4 \)
- This is a difference of squares: \( a^2 - b^2 = (a - b)(a + b) \).
- Here, \( 9x^2 = (3x)^2 \) and \( 4 = 2^2 \).
- Factor: \( (3x - 2)(3x + 2) \).

Answer: \( (3x - 2)(3x + 2) \)

#### 15. \( 9x^2 - 12x + 4 \)
- This is a perfect square trinomial: \( a^2 - 2ab + b^2 = (a - b)^2 \).
- Here, \( 9x^2 = (3x)^2 \), \( -12x = -2 \cdot 3x \cdot 2 \), and \( 4 = 2^2 \).
- Factor: \( (3x - 2)^2 \).

Answer: \( (3x - 2)^2 \)

#### 16. \( 9x^2 + 12x + 4 \)
- This is a perfect square trinomial: \( a^2 + 2ab + b^2 = (a + b)^2 \).
- Here, \( 9x^2 = (3x)^2 \), \( 12x = 2 \cdot 3x \cdot 2 \), and \( 4 = 2^2 \).
- Factor: \( (3x + 2)^2 \).

Answer: \( (3x + 2)^2 \)

#### 17. \( x^2 + 8x - 9 \)
- This is a trinomial.
- We need two numbers that multiply to \( 1 \cdot (-9) = -9 \) and add to \( 8 \).
- The numbers are \( 9 \) and \( -1 \).
- Rewrite the middle term: \( x^2 + 9x - x - 9 \).
- Group and factor: \( (x^2 + 9x) - (x + 9) = x(x + 9) - 1(x + 9) \).
- Factor out the common binomial: \( (x + 9)(x - 1) \).

Answer: \( (x + 9)(x - 1) \)

#### 18. \( x^2 - 10x + 9 \)
- This is a trinomial.
- We need two numbers that multiply to \( 1 \cdot 9 = 9 \) and add to \( -10 \).
- The numbers are \( -9 \) and \( -1 \).
- Rewrite the middle term: \( x^2 - 9x - x + 9 \).
- Group and factor: \( (x^2 - 9x) - (x - 9) = x(x - 9) - 1(x - 9) \).
- Factor out the common binomial: \( (x - 9)(x - 1) \).

Answer: \( (x - 9)(x - 1) \)

#### 19. \( x^2 - 8x - 9 \)
- This is a trinomial.
- We need two numbers that multiply to \( 1 \cdot (-9) = -9 \) and add to \( -8 \).
- The numbers are \( -9 \) and \( 1 \).
- Rewrite the middle term: \( x^2 - 9x + x - 9 \).
- Group and factor: \( (x^2 - 9x) + (x - 9) = x(x - 9) + 1(x - 9) \).
- Factor out the common binomial: \( (x - 9)(x + 1) \).

Answer: \( (x - 9)(x + 1) \)

#### 20. \( x^2 - 9 \)
- This is a difference of squares: \( a^2 - b^2 = (a - b)(a + b) \).
- Here, \( x^2 = x^2 \) and \( 9 = 3^2 \).
- Factor: \( (x - 3)(x + 3) \).

Answer: \( (x - 3)(x + 3) \)

#### 21. \( x^2 - 6x + 9 \)
- This is a perfect square trinomial: \( a^2 - 2ab + b^2 = (a - b)^2 \).
- Here, \( x^2 = x^2 \), \( -6x = -2 \cdot x \cdot 3 \), and \( 9 = 3^2 \).
- Factor: \( (x - 3)^2 \).

Answer: \( (x - 3)^2 \)

#### 22. \( x^2 + 6x + 9 \)
- This is a perfect square trinomial: \( a^2 + 2ab + b^2 = (a + b)^2 \).
- Here, \( x^2 = x^2 \), \( 6x = 2 \cdot x \cdot 3 \), and \( 9 = 3^2 \).
- Factor: \( (x + 3)^2 \).

Answer: \( (x + 3)^2 \)

#### 23. \( 2x^2 + 7x + 5 \)
- This is a trinomial.
- We need two numbers that multiply to \( 2 \cdot 5 = 10 \) and add to \( 7 \).
- The numbers are \( 5 \) and \( 2 \).
- Rewrite the middle term: \( 2x^2 + 5x + 2x + 5 \).
- Group and factor: \( (2x^2 + 5x) + (2x + 5) = x(2x + 5) + 1(2x + 5) \).
- Factor out the common binomial: \( (2x + 5)(x + 1) \).

Answer: \( (2x + 5)(x + 1) \)

#### 24. \( 2x^2 + 3x - 5 \)
- This is a trinomial.
- We need two numbers that multiply to \( 2 \cdot (-5) = -10 \) and add to \( 3 \).
- The numbers are \( 5 \) and \( -2 \).
- Rewrite the middle term: \( 2x^2 + 5x - 2x - 5 \).
- Group and factor: \( (2x^2 + 5x) - (2x + 5) = x(2x + 5) - 1(2x + 5) \).
- Factor out the common binomial: \( (2x + 5)(x - 1) \).

Answer: \( (2x + 5)(x - 1) \)

#### 25. \( 2x^2 - 7x + 5 \)
- This is a trinomial.
- We need two numbers that multiply to \( 2 \cdot 5 = 10 \) and add to \( -7 \).
- The numbers are \( -5 \) and \( -2 \).
- Rewrite the middle term: \( 2x^2 - 5x - 2x + 5 \).
- Group and factor: \( (2x^2 - 5x) - (2x - 5) = x(2x - 5) - 1(2x - 5) \).
- Factor out the common binomial: \( (2x - 5)(x - 1) \).

Answer: \( (2x - 5)(x - 1) \)

#### 26. \( 2x^2 - 9x - 5 \)
- This is a trinomial.
- We need two numbers that multiply to \( 2 \cdot (-5) = -10 \) and add to \( -9 \).
- The numbers are \( -10 \) and \( 1 \).
- Rewrite the middle term: \( 2x^2 - 10x + x - 5 \).
- Group and factor: \( (2x^2 - 10x) + (x - 5) = 2x(x - 5) + 1(x - 5) \).
- Factor out the common binomial: \( (2x + 1)(x - 5) \).

Answer: \( (2x + 1)(x - 5) \)

#### 27. \( 2x^2 - 11x + 5 \)
- This is a trinomial.
- We need two numbers that multiply to \( 2 \cdot 5 = 10 \) and add to \( -11 \).
- The numbers are \( -10 \) and \( -1 \).
- Rewrite the middle term: \( 2x^2 - 10x - x + 5 \).
- Group and factor: \( (2x^2 - 10x) - (x - 5) = 2x(x - 5) - 1(x - 5) \).
- Factor out the common binomial: \( (2x - 1)(x - 5) \).

Answer: \( (2x - 1)(x - 5) \)

#### 28. \( 2x^2 + 9x - 5 \)
- This is a trinomial.
- We need two numbers that multiply to \( 2 \cdot (-5) = -10 \) and add to \( 9 \).
- The numbers are \( 10 \) and \( -1 \).
- Rewrite the middle term: \( 2x^2 + 10x - x - 5 \).
- Group and factor: \( (2x^2 + 10x) - (x + 5) = 2x(x + 5) - 1(x + 5) \).
- Factor out the common binomial: \( (2x - 1)(x + 5) \).

Answer: \( (2x - 1)(x + 5) \)

#### 29. \( x^2 - 16 \)
- This is a difference of squares: \( a^2 - b^2 = (a - b)(a + b) \).
- Here, \( x^2 = x^2 \) and \( 16 = 4^2 \).
- Factor: \( (x - 4)(x + 4) \).

Answer: \( (x - 4)(x + 4) \)

#### 30. \( x^2 + 8x + 16 \)
- This is a perfect square trinomial: \( a^2 + 2ab + b^2 = (a + b)^2 \).
- Here, \( x^2 = x^2 \), \( 8x = 2 \cdot x \cdot 4 \), and \( 16 = 4^2 \).
- Factor: \( (x + 4)^2 \).

Answer: \( (x + 4)^2 \)

#### 31. \( x^2 - 8x + 16 \)
- This is a perfect square trinomial: \( a^2 - 2ab + b^2 = (a - b)^2 \).
- Here, \( x^2 = x^2 \), \( -8x = -2 \cdot x \cdot 4 \), and \( 16 = 4^2 \).
- Factor: \( (x - 4)^2 \).

Answer: \( (x - 4)^2 \)

#### 32. \( 2x^2 - 16x + 32 \)
- First, factor out the GCF, which is \( 2 \): \( 2(x^2 - 8x + 16) \).
- The expression inside the parentheses is a perfect square trinomial: \( x^2 - 8x + 16 = (x - 4)^2 \).
- Factor: \( 2(x - 4)^2 \).

Answer: \( 2(x - 4)^2 \)

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Final Answer:

\[
\boxed{
\begin{aligned}
1. & \ (5x + 3)(x + 1) \\
2. & \ (5x + 3)(x - 1) \\
3. & \ (5x - 3)(x - 1) \\
4. & \ (5x - 3)(x + 1) \\
5. & \ (x + 7)(x - 1) \\
6. & \ (x - 7)(x + 1) \\
7. & \ (x - 7)(x - 1) \\
8. & \ (7x + 3)(x + 1) \\
9. & \ (7x - 3)(x - 1) \\
10. & \ (7x - 3)(x + 1) \\
11. & \ (7x + 1)(x + 3) \\
12. & \ (7x - 1)(x + 3) \\
13. & \ (7x - 1)(x - 3) \\
14. & \ (3x - 2)(3x + 2) \\
15. & \ (3x - 2)^2 \\
16. & \ (3x + 2)^2 \\
17. & \ (x + 9)(x - 1) \\
18. & \ (x - 9)(x - 1) \\
19. & \ (x - 9)(x + 1) \\
20. & \ (x - 3)(x + 3) \\
21. & \ (x - 3)^2 \\
22. & \ (x + 3)^2 \\
23. & \ (2x + 5)(x + 1) \\
24. & \ (2x + 5)(x - 1) \\
25. & \ (2x - 5)(x - 1) \\
26. & \ (2x + 1)(x - 5) \\
27. & \ (2x - 1)(x - 5) \\
28. & \ (2x - 1)(x + 5) \\
29. & \ (x - 4)(x + 4) \\
30. & \ (x + 4)^2 \\
31. & \ (x - 4)^2 \\
32. & \ 2(x - 4)^2
\end{aligned}
}
\]