To fully factorize the quadratic expression \(6x^2 + 9x - 10\), follow these detailed steps:
1. Identify coefficients:
- \(a = 6\)
- \(b = 9\)
- \(c = -10\)
2. Multiply \(a\) and \(c\):
- \(ac = 6 \times (-10) = -60\)
3. Find two numbers that multiply to \(ac\) and add to \(b\):
- We need two numbers that multiply to \(-60\) and add to \(9\).
These numbers are \(15\) and \(-4\):
- \(15 \times (-4) = -60\)
- \(15 + (-4) = 9\)
4. Rewrite the middle term \(9x\) using \(15\) and \(-4\):
- \(6x^2 + 15x - 4x - 10\)
5. Factor by grouping:
- Group the terms:
[tex]\[(6x^2 + 15x) + (-4x - 10)\][/tex]
- Factor out the greatest common factor (GCF) from each group:
[tex]\[3x(2x + 5) - 2(2x + 5)\][/tex]
6. Factor out the common binomial \((2x + 5)\):
- \((2x + 5)(3x - 2)\)
So, the fully factorized form of the expression \(6x^2 + 9x - 10\) is:
[tex]\[
(2x + 5)(3x - 2)
\][/tex]
