Answer
$x=\left\{ -4,-\dfrac{2}{3} \right\}$
Work Step by Step
Using $(a+b)(c+d)=ac+ad+bc+bd$ or the Distributive Property, the given expression, $
(x+3)(3x+5)=7
,$ is equivalent to
\begin{array}{l}\require{cancel}
x(3x)+x(5)+3(3x)+3(5)=7
\\\\
3x^2+5x+9x+15=7
\\\\
3x^2+5x+9x+15-7=0
\\\\
3x^2+14x+8=0
.\end{array}
Factoring the above equation, $
3x^2+14x+8=0
,$ results to
\begin{array}{l}\require{cancel}
(x+4)(3x+2)=0
.\end{array}
Equating each factor to zero (Zero Product Principle), then the solutions to the equation, $
(x+4)(3x+2)=0
,$ are
\begin{array}{l}\require{cancel}
x+4=0
\\\\
x=-4
,\\\\\text{OR}\\\\
3x+2=0
\\\\
3x=-2
\\\\
x=-\dfrac{2}{3}
.\end{array}
Hence, $
x=\left\{ -4,-\dfrac{2}{3} \right\}
.$