The general term in the binomial expansion of (a+b)ⁿ is given by the formula:

The correct option is B.
The fourth term is $\boxed{\text{B. } -84x^6y^3}$.

: Obtain the fourth term in the expansion of $(x^2 + 3)^6$.
Here, $a=x^2$, $b=3$, $n=6$. For the fourth term, $k+1=4$, so $k=3$.

The correct option is B.
The fourth term is $\boxed{\text{B. } 540x^6}$.

: Obtain the sixth term in the expansion of $(3x - 2y)^9$.
Here, $a=3x$, $b=-2y$, $n=9$. For the sixth term, $k+1=6$, so $k=5$.

The correct option is D.
The sixth term is $\boxed{\text{D. } -326592x^4y^5}$.

: Obtain the second term in the expansion of $(2x - 1)^2$.
Here, $a=2x$, $b=-1$, $n=2$. For the second term, $k+1=2$, so $k=1$.

The correct option is C.
The second term is $\boxed{\text{C. } -4x}$.

: What is the coefficient of $x^3$ in $(2 + 3x)^4$?
Here, $a=2$, $b=3x$, $n=4$. We need to find the term where the power of $x$ is 3.
The general term is $T_{k+1} = \binom{n}{k} a^{n-k} b^k$.

For the term containing $x^3$, we must have $x^k = x^3$, so $k=3$.

The coefficient of $x^3$ is 216.
The correct option is D.
The coefficient of $x^3$ is $\boxed{\text{D. } 216}$.