To factor the quadratic expression [tex]\( 9x^2 + 3x - 2 \)[/tex], we need to find two binomials whose product gives us this polynomial. Here are the steps to solve this problem.
1. Write down the polynomial:
[tex]\[ 9x^2 + 3x - 2 \][/tex]
2. Identify the terms in the polynomial:
- The coefficient of [tex]\( x^2 \)[/tex] is 9.
- The coefficient of [tex]\( x \)[/tex] is 3.
- The constant term is -2.
3. Find two numbers that multiply to give the product of the leading coefficient (9) and the constant term (-2).
The product of 9 and -2 is:
[tex]\[ 9 \cdot (-2) = -18 \][/tex]
4. We need two numbers that multiply to -18 and add up to the middle coefficient, 3. These numbers are 6 and -3 because:
[tex]\[ 6 \times (-3) = -18 \][/tex]
[tex]\[ 6 + (-3) = 3 \][/tex]
5. Rewrite the middle term using the two numbers found:
[tex]\[ 9x^2 + 6x - 3x - 2 \][/tex]
6. Factor by grouping:
Group the terms to factor out the common factors:
[tex]\[ (9x^2 + 6x) + (-3x - 2) \][/tex]
Factor out the greatest common factor from each group:
[tex]\[ 3x(3x + 2) - 1(3x + 2) \][/tex]
7. Factor out the common binomial factor:
Notice that [tex]\( 3x + 2 \)[/tex] is a common factor:
[tex]\[ (3x + 2)(3x - 1) \][/tex]
Therefore, the factors of the polynomial [tex]\( 9x^2 + 3x - 2 \)[/tex] are:
[tex]\[ (3x + 2)(3x - 1) \][/tex]
So, the correct answer is:
C. [tex]\( (3x + 2)(3x - 1) \)[/tex]
