Answer
$\left( 3m^{\frac{1}{2}}-2 \right)\left(m^{\frac{1}{2}} - 7 \right)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
3m-23m^{\frac{1}{2}}+14
,$ 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 $
3(14)=42
$ and the value of $b$ is $
-23
.$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{
-2,-21
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
3m-2m^{\frac{1}{2}}-21m^{\frac{1}{2}}+14
.\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}
\left( 3m-2m^{\frac{1}{2}} \right) - \left( 21m^{\frac{1}{2}}-14 \right)
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
m^{\frac{1}{2}}\left( 3m^{\frac{1}{2}}-2 \right) - 7\left( 3m^{\frac{1}{2}}-2 \right)
.\end{array}
Factoring the $GCF=
\left( 3m^{\frac{1}{2}}-2 \right)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
\left( 3m^{\frac{1}{2}}-2 \right)\left(m^{\frac{1}{2}} - 7 \right)
.\end{array}
