To factor the quadratic expression [tex]\(5x^2 - 17x - 12\)[/tex] completely, let us identify the factors step by step.
1. Identify the quadratic expression to be factored:
[tex]\[
5x^2 - 17x - 12
\][/tex]
2. Express your factored form as:
[tex]\[
(x - a)(bx + c)
\][/tex]
Here, we need to find the correct values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]. For our specific problem, we will simplify the steps.
3. Identify the expression in the form:
[tex]\[
(x - \alpha)(5x + \beta)
\][/tex]
where [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex] are constants we need to determine.
4. Knowing the final form,
[tex]\[
(x - 4)(5x + 3)
\][/tex]
Plugging in these values, we see the factored form is:
[tex]\[
\boxed{(x - 4)(5x + 3)}
\][/tex]
By examining this factorization, we verify that the expression indeed becomes:
[tex]\[
(x - 4)(5x + 3)
\][/tex]
Therefore, the given expression [tex]\(5x^2 - 17x - 12\)[/tex] factors to:
[tex]\[
(x - 4)(5x + 3)
\][/tex]
