Answer
$(r-6p)(r+7p)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
r^2+rp-42p^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 expression above, the value of $c$ is $
-42
$ and the value of $b$ is $
1
.$
The possible pairs of integers whose product is $c$ are
\begin{array}{l}\require{cancel}
\{ 1,-42 \}, \{ 2,-21 \}, \{ 3,-14 \}, \{ 6,-7 \},
\\
\{ -1,42 \}, \{ -2,21 \}, \{ -3,14 \}, \{ -6,7 \}
.\end{array}
Among these pairs, the one that gives a sum of $b$ is $\{
-6,7
\}.$ Hence, the factored form of the expression above is
\begin{array}{l}\require{cancel}
(r-6p)(r+7p)
.\end{array}
