Answer
$(m-5n)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
m^2-3mn-10n^2
,$ find two numbers, $m_1$ and $m_2,$ whose product is $c$ and whose sum is $b$ in the quadratic expression $x^2+bx+c.$ Then, express the factored form as $(x+m_1)(x+m_2).$
$\bf{\text{Solution Details:}}$
In the given expression, the value of $c$ is $
-10
$ and the value of $b$ is $
-3
.$
The possible pairs of integers whose product is $c$ are
\begin{array}{l}\require{cancel}
\{ 1,-10 \}, \{ 2,-5 \},
\{ -1,10 \}, \{ -2,5 \}
.\end{array}
Among these pairs, the one that gives a sum of $b$ is $\{
2,-5
\}.$ Hence, the factored form of the given expression is
\begin{array}{l}\require{cancel}
(m+2n)(m-5n)
.\end{array}
The missing factor in the given expression is $
(m-5n)
.$