Answer
$(2k-h)(5k-3h)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
10k^2-11kh+3h^2
,$ 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.
$\bf{\text{Solution Details:}}$
To factor the trinomial expression above, note that the value of $ac$ is $
10(3)=30
$ and the value of $b$ is $
-11
.$
The possible pairs of integers whose product is $ac$ are
\begin{array}{l}\require{cancel}
\{1,30\}, \{2,15\}, \{3,10\}, \{5,6\},
\\
\{-1,-30\}, \{-2,-15\}, \{-3,-10\}, \{-5,-6\}
.\end{array}
Among these pairs, the one that gives a sum of $b$ is $\{
-5,-6
\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
10k^2-5kh-6kh+3h^2
.\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}
(10k^2-5kh)-(6kh-3h^2)
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
5k(2k-h)-3h(2k-h)
.\end{array}
Factoring the $GCF=
(2k-h)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(2k-h)(5k-3h)
.\end{array}