Let's factor each quadratic trinomial one by one.

Part (a) $x^2 - 4x - 12$

1. The expression is $x^2 - 4x - 12$.
2. We need to find two numbers that multiply to $-12$ and add up to $-4$.
   • These numbers are $-6$ and $2$.
3. Rewrite the middle term using these numbers: $x^2 - 6x + 2x - 12$
4. Factor by grouping: $= x(x - 6) + 2(x - 6)$
5. Factor out the common term: $= (x + 2)(x - 6)$

Solution for (a): $(x + 2)(x - 6)$

Part (b) $x^2 - 11x + 18$

1. The expression is $x^2 - 11x + 18$.
2. Find two numbers that multiply to $18$ and add up to $-11$.
   • These numbers are $-9$ and $-2$.
3. Rewrite the middle term using these numbers: $x^2 - 9x - 2x + 18$
4. Factor by grouping: $= x(x - 9) - 2(x - 9)$
5. Factor out the common term: $= (x - 2)(x - 9)$

Solution for (b): $(x - 2)(x - 9)$

Part (c) $6x^2 - 13x + 6$

1. The expression is $6x^2 - 13x + 6$.
2. Find two numbers that multiply to $6 \times 6 = 36$ and add up to $-13$.
   • These numbers are $-9$ and $-4$.
3. Rewrite the middle term using these numbers: $6x^2 - 9x - 4x + 6$
4. Factor by grouping: $= 3x(2x - 3) - 2(2x - 3)$
5. Factor out the common term: $= (3x - 2)(2x - 3)$

Solution for (c): $(3x - 2)(2x - 3)$

Would you like more details or have any questions on any step?

Related Questions

1. How can we factor trinomials where the leading coefficient is not 1?
2. What are some common mistakes to avoid when factoring trinomials?
3. How can the quadratic formula help in factoring trinomials?
4. When is it appropriate to use the grouping method in factoring?
5. How do you verify if the factorization of a trinomial is correct?

Tip

Always check your factors by expanding them back out to ensure you get the original trinomial. This verifies the accuracy of your factorization.