Answer
The value of $k$ is $-12$
Work Step by Step
If $x-c$ is factor of $f\left( x \right)$, then $f\left( c \right)=0$
so, for $4x+3$ to be factor of $f\left( x \right)$:
$f\left( \frac{-3}{4} \right)=0$.
Thus,
$\begin{align}
& 20{{\left( \frac{-3}{4} \right)}^{3}}+23{{\left( \frac{-3}{4} \right)}^{2}}-10\left( \frac{-3}{4} \right)+k=0 \\
& \frac{-135}{16}+\frac{207}{16}+\frac{15}{2}+k=0 \\
& k=\frac{135-207-120}{16} \\
& k=\frac{-192}{16}=-12
\end{align}$
Therefore, the value of $k$ is $-12$.
