Answer
$4p-36\sqrt{p}+\sqrt{7p}-9\sqrt{7}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given radical expression, $
(4\sqrt{p}+\sqrt{7})(\sqrt{p}-9)
,$ use FOIL and the laws of radicals.
$\bf{\text{Solution Details:}}$
Using the FOIL Method which is given by $(a+b)(c+d)=ac+ad+bc+bd,$ the expression above is equivalent to\begin{array}{l}\require{cancel}
4\sqrt{p}(\sqrt{p})+4\sqrt{p}(-9)+\sqrt{7}(\sqrt{p})+\sqrt{7}(-9)
\\\\=
4(\sqrt{p})^2+4(-9)\sqrt{p}+\sqrt{7}(\sqrt{p})+1(-9)\sqrt{7}
\\\\=
4p-36\sqrt{p}+\sqrt{7}(\sqrt{p})-9\sqrt{7}
.\end{array}
Using the Product Rule of radicals which is given by $\sqrt[m]{x}\cdot\sqrt[m]{y}=\sqrt[m]{xy},$ the expression above is equivalent to\begin{array}{l}\require{cancel}
4p-36\sqrt{p}+\sqrt{7(p)}-9\sqrt{7}
\\\\=
4p-36\sqrt{p}+\sqrt{7p}-9\sqrt{7}
.\end{array}
