Answer
$p=\left\{ -4,0 \right\}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Express the given equation, $
4p^2+16p=0
,$ in factored form. Then, use the Zero-Factor Property by equating each factor to zero. Finally, solve each equation.
$\bf{\text{Solution Details:}}$
The factored form of the equation above is
\begin{array}{l}\require{cancel}
4p(p+4)=0
.\end{array}
Equating each factor to zero (Zero-Factor Property), then
\begin{array}{l}\require{cancel}
4p=0
\text{ OR }
p+4=0
.\end{array}
Using the properties of equality to solve each of the equation above results to
\begin{array}{l}\require{cancel}
4p=0
\\\\
p=\dfrac{0}{4}
\\\\
p=0
\\\\\text{ OR }\\\\
p+4=0
\\\\
p=-4
.\end{array}
Hence, the solutions are $
p=\left\{ -4,0 \right\}
.$
$\bf{\text{Supplementary Solution/s:}}$
In the expression $
4p^2+16p
,$ the $GCF$ of the constants of the terms $\{
4, 16
\}$ is $
4
.$ The $GCF$ of the common variable/s is the variable/s with the lowest exponent. Hence, the $GCF$ of the common variable/s $\{
p^2,p
\}$ is $
p
.$ Hence, the entire expression has $GCF=
4p
.$
Factoring the $GCF=
4p
,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
4p \left( \dfrac{4p^2}{4p}+\dfrac{16p}{4p} \right)
.\end{array}
Using the Quotient Rule of the laws of exponents which states that $\dfrac{x^m}{x^n}=x^{m-n},$ the expression above simplifies to
\begin{array}{l}\require{cancel}
4p \left( p^{2-1}+4p^{1-1} \right)
\\\\=
4p \left( p^{1}+4p^{0} \right)
\\\\=
4p \left( p+4(1) \right)
\\\\=
4p \left( p+4 \right)
.\end{array}
