To factor trinomials, especially those of the form ax2+bx+cax^2+bx+cax2+bx+c, follow these general steps:

Factoring Trinomials When a=1a=1a=1 (Leading Coefficient is 1)

1. Identify bbb and ccc in the trinomial x2+bx+cx^2+bx+cx2+bx+c.
2. Find two numbers that multiply to ccc and add to bbb.
3. Write the factors as (x+m)(x+n)(x+m)(x+n)(x+m)(x+n), where mmm and nnn are the two numbers found.
4. Check your work by expanding the binomials to ensure it equals the original trinomial.

Example: Factor x2+5x+6x^2+5x+6x2+5x+6.

• Numbers that multiply to 6 and add to 5 are 2 and 3.
• Factors: (x+2)(x+3)(x+2)(x+3)(x+2)(x+3)

Factoring Trinomials When a≠1a\neq 1a=1

Use the AC method (also called the grouping method):

1. Multiply aaa and ccc (the coefficient of x2x^2x2 and the constant term).
2. Find two numbers that multiply to acacac and add to bbb.
3. Rewrite the middle term bxbxbx as the sum of two terms using the two numbers found.
4. Factor by grouping : group terms in pairs and factor out the greatest common factor (GCF) from each group.
5. Factor out the common binomial factor from the two groups.
6. Check by multiplying the factors to confirm the original trinomial.

Example: Factor 18x2−31x+618x^2-31x+618x2−31x+6.

• a×c=18×6=108a\times c=18\times 6=108a×c=18×6=108.
• Find two numbers that multiply to 108 and add to -31: -4 and -27.
• Rewrite: 18x2−4x−27x+618x^2-4x-27x+618x2−4x−27x+6.
• Group: (18x2−4x)+(−27x+6)(18x^2-4x)+(-27x+6)(18x2−4x)+(−27x+6).
• Factor each group: 2x(9x−2)−3(9x−2)2x(9x-2)-3(9x-2)2x(9x−2)−3(9x−2).
• Factor out common binomial: (2x−3)(9x−2)(2x-3)(9x-2)(2x−3)(9x−2)

Summary of Key Points

• For a=1a=1a=1, find two numbers that multiply to ccc and add to bbb.
• For a≠1a\neq 1a=1, use the AC method involving factoring by grouping.
• Always check your factors by expanding to verify correctness.

This method works reliably for factoring most trinomials encountered in algebra