To factor the given trinomial \( 12x^2 - 7x - 5 \), we will follow a process of trial and error, observation, or using techniques like the AC method. However, let's directly go to the step-by-step solution:
1. Recognize the Trinomial: We need to factor \( 12x^2 - 7x - 5 \).
2. Find the constant and the coefficients:
- The coefficient of \(x^2\) (the leading coefficient) is 12.
- The coefficient of \(x\) is -7.
- The constant term is -5.
3. Set Up Factoring by Grouping:
- We are looking for two binomials whose product is \( 12x^2 - 7x - 5 \).
4. Factorization:
- We can write the trinomial in a factored form:
[tex]\[
12x^2 - 7x - 5 = (ax + b)(cx + d)
\][/tex]
5. Matching the Coefficients:
- After expanding \( (ax + b)(cx + d) \), you get \( acx^2 + (ad + bc)x + bd \).
- We need to find \(a\), \(b\), \(c\), and \(d\) that satisfy \( ac = 12 \), \( bd = -5 \), and \( ad + bc = -7\).
6. Using the Result:
- By analyzing the patterns, we find:
[tex]\[
(x - 1)(12x + 5)
\][/tex]
7. Verification:
- We can expand to verify the result:
[tex]\[
(x - 1)(12x + 5) = x \cdot 12x + x \cdot 5 - 1 \cdot 12x - 1 \cdot 5 = 12x^2 + 5x - 12x - 5 = 12x^2 - 7x - 5
\][/tex]
So, the factorization of the trinomial \( 12x^2 - 7x - 5 \) is:
[tex]\[
\boxed{(x - 1)(12x + 5)}
\][/tex]
Hence, the given trinomial [tex]\( 12x^2 - 7x - 5 \)[/tex] is factored as [tex]\( (x - 1)(12x + 5) \)[/tex].
