1. Find vertex $A$ from the intersection of the given sides

The sides $AB$ and $AC$ are: $4x+y=14$ $3x-2y=5$

Solve simultaneously.

From the first equation, $y=14-4x$

Substitute into the second: $3x-2(14-4x)=5$ $3x-28+8x=5$ $11x=33$ $x=3$

Then, $y=14-4(3)=2$

So, $A=(3,2)$

2. Use the section formula on side $BC$

Given that the point $P=\left(2,-\frac{4}{3}\right)$ divides side $BC$ internally in the ratio $2:1$.

So if $B=(x_1,y_1), \quad C=(x_2,y_2),$ then $P=\left(\frac{2x_2+x_1}{3},\frac{2y_2+y_1}{3}\right)$ or equivalently, since $BP:PC=2:1$, $P=\left(\frac{2x_C+x_B}{3},\frac{2y_C+y_B}{3}\right)$

But here $B$ lies on line $AB$, $C$ lies on line $AC$, and $A,B,C$ form a triangle.

A simpler way: since $P$ divides $BC$ in the ratio $2:1$, we can express $P=\frac{B+2C}{3}$ So, $3P=B+2C$

Let us write points $B$ and $C$ on the two given lines through $A=(3,2)$.

3. Parametrize points on $AB$ and $AC$

Line $AB: 4x+y=14$ has slope $-4$, so a direction vector is $(1,-4)$. Hence any point on $AB$ can be written as $B=(3,2)+s(1,-4)=(3+s,2-4s)$

Line $AC: 3x-2y=5$ has slope $\frac{3}{2}$, so a direction vector is $(2,3)$. Hence any point on $AC$ can be written as $C=(3,2)+t(2,3)=(3+2t,2+3t)$

Now use $P=\left(2,-\frac{4}{3}\right)=\frac{B+2C}{3}$

So, $B+2C=(6,-4)$

Substitute $B$ and $C$: $B+2C=(3+s,2-4s)+2(3+2t,2+3t)$ $=(3+s,2-4s)+(6+4t,4+6t)$ $=(9+s+4t,6-4s+6t)$

Equate coordinates: $9+s+4t=6$ $6-4s+6t=-4$

Thus, $s+4t=-3 \quad ...(1)$ $-4s+6t=-10 \quad ...(2)$

From (1), $s=-3-4t$

Substitute into (2): $-4(-3-4t)+6t=-10$ $12+16t+6t=-10$ $22t=-22$ $t=-1$

Then, $s=-3-4(-1)=1$

So, $B=(3+1,2-4)=(4,-2)$ $C=(3+2(-1),2+3(-1))=(1,-1)$

4. Find equation of line $BC$ through $B(4,-2)$ and $C(1,-1)$

Slope of $BC$ is $m=\frac{-1-(-2)}{1-4}=\frac{1}{-3}=-\frac13$

Using point-slope form through $(4,-2)$: $y+2=-\frac13(x-4)$

Multiply by $3$: $3y+6=-x+4$ $x+3y+2=0$

5. Match with the options

The equation is $\boxed{x+3y+2=0}$ which is Option C.

6. Comparison with stored correct answer

Stored correct answer: C

Our derived answer: C

So the derived answer agrees with the stored correct answer.