$6{x}^{2}+10x+3x+5=0$
Factor out $2x$ from the expression$2x \times \left( 3x+5 \right)+3x+5=0$
Factor out $3x+5$ from the expression$\left( 3x+5 \right) \times \left( 2x+1 \right)=0$
When the product of factors equals $0$, at least one factor is $0$$\begin{array} { l }3x+5=0,\\2x+1=0\end{array}$
Solve the equation for $x$$\begin{array} { l }x=-\frac{ 5 }{ 3 },\\2x+1=0\end{array}$
Solve the equation for $x$$\begin{array} { l }x=-\frac{ 5 }{ 3 },\\x=-\frac{ 1 }{ 2 }\end{array}$
The equation has $2$ solutions$\begin{align*}&\begin{array} { l }x_1=-\frac{ 5 }{ 3 },& x_2=-\frac{ 1 }{ 2 }\end{array} \\&\begin{array} { l }x_1\approx-1.66667,& x_2=-0.5\end{array}\end{align*}$