To factor the quadratic expression [tex]\(3x^2 - 11x - 20\)[/tex], let's follow these steps:
1. Identify the coefficients:
- [tex]\(a = 3\)[/tex] (coefficient of [tex]\(x^2\)[/tex])
- [tex]\(b = -11\)[/tex] (coefficient of [tex]\(x\)[/tex])
- [tex]\(c = -20\)[/tex] (constant term)
2. Set up the factors in the form:
[tex]\[
(Ax + B)(Cx + D)
\][/tex]
where [tex]\(A \times C = a\)[/tex] and [tex]\(B \times D = c\)[/tex].
3. Determine the correct pairs:
- Look for pairs of factors of [tex]\(ac\)[/tex] that add up to [tex]\(b\)[/tex]. Here, [tex]\(ac = 3 \times (-20) = -60\)[/tex] and [tex]\(b = -11\)[/tex].
4. Find the correct pair:
- The pair of numbers that multiply to [tex]\(-60\)[/tex] and add up to [tex]\(-11\)[/tex] is [tex]\(4\)[/tex] and [tex]\(-15\)[/tex].
5. Rewrite the middle term:
[tex]\[
3x^2 - 15x + 4x - 20
\][/tex]
6. Group the terms and factor each group:
[tex]\[
(3x^2 - 15x) + (4x - 20)
\][/tex]
[tex]\[
= 3x(x - 5) + 4(x - 5)
\][/tex]
7. Factor out the common binomial factor:
[tex]\[
= (3x + 4)(x - 5)
\][/tex]
Thus, the factored form of the quadratic expression [tex]\(3x^2 - 11x - 20\)[/tex] is:
[tex]\[
(3x + 4)(x - 5)
\][/tex]
So the correct answer is:
[tex]\[
(3x + \boxed{4})(x - \boxed{5})
\][/tex]
