Answer
$(5b+1)(b-3)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
5b^2-14b-3
,$ find two numbers whose product is $ac$ and whose sum is $b$ in the quadratic expression $ax^2+bx+c.$ Use these $2$ numbers to decompose the middle term of the given quadratic expression and then use factoring by grouping.
$\bf{\text{Solution Details:}}$
Using factoring of trinomials, the value of $ac$ in the trinomial expression above is $
5(-3)=-15
$ and the value of $b$ is $
-14
.$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{
1,-15
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
5b^2+b-15b-3
.\end{array}
Grouping the first and second terms and the third and fourth terms, the given expression is equivalent to
\begin{array}{l}\require{cancel}
(5b^2+b)-(15b+3)
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
b(5b+1)-3(5b+1)
.\end{array}
Factoring the $GCF=
(5b+1)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(5b+1)(b-3)
.\end{array}
