Answer
$x=\left\{ -\dfrac{8}{3},0 \right\}$
Work Step by Step
Changing to exponential form, the given expression, $
\log_5[(3x+5)(x+1)]=1
,$
is equivalent to
\begin{array}{l}\require{cancel}
(3x+5)(x+1)=5^1
.\end{array}
Using concepts of quadratic equations, the solution/s to the equation above is/are
\begin{array}{l}\require{cancel}
3x(x)+3x(1)+5(x)+5(1)=5
\\\\
3x^2+3x+5x+5-5=0
\\\\
3x^2+8x=0
\\\\
x(3x+8)=0
.\end{array}
Equating each factor to zero and then solving for the variable, the solutions are $
x=\left\{ -\dfrac{8}{3},0 \right\}
.$
Upon checking, $
x=\left\{ -\dfrac{8}{3},0 \right\}
$ satisfy the original equation.
