Sure! Let's factor the quadratic polynomial [tex]\(6x^2 + 19x + 10\)[/tex] step-by-step.
1. Identify the coefficients:
- The quadratic term coefficient [tex]\(a\)[/tex] is 6.
- The linear term coefficient [tex]\(b\)[/tex] is 19.
- The constant term [tex]\(c\)[/tex] is 10.
2. Look for two numbers that multiply to [tex]\(a \times c\)[/tex] (which is 6 \times 10 = 60), and add up to [tex]\(b\)[/tex] (which is 19).
These two numbers are 15 and 4 because:
- [tex]\(15 \times 4 = 60\)[/tex]
- [tex]\(15 + 4 = 19\)[/tex]
3. Rewrite the middle term (19x) using these two numbers:
[tex]\[
6x^2 + 19x + 10 = 6x^2 + 15x + 4x + 10
\][/tex]
4. Group the terms in pairs:
[tex]\[
(6x^2 + 15x) + (4x + 10)
\][/tex]
5. Factor out the greatest common factor from each pair:
[tex]\[
3x(2x + 5) + 2(2x + 5)
\][/tex]
6. Notice that [tex]\(2x + 5\)[/tex] is a common factor:
[tex]\[
(3x + 2)(2x + 5)
\][/tex]
So, the factored form of the given polynomial [tex]\(6x^2 + 19x + 10\)[/tex] is:
[tex]\[
(3x + 2)(2x + 5)
\][/tex]
