## Precalculus (6th Edition) Blitzer

The value of $k$ is $-12$
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 $-12$.