use the long - division method to find the result when (4x^{3}+4x^{2}-19x - 24) is divided by (2x + 3).

Explanation:

Step1: Set up long - division

Divide the first term of the dividend $4x^{3}+4x^{2}+0x - 24$ (inserted $0x$ for the missing $x$ - term) by the first term of the divisor $2x+3$. $\frac{4x^{3}}{2x}=2x^{2}$.

Step2: Multiply and subtract

Multiply $2x + 3$ by $2x^{2}$ to get $4x^{3}+6x^{2}$. Subtract this from the dividend: $(4x^{3}+4x^{2}+0x - 24)-(4x^{3}+6x^{2})=-2x^{2}+0x - 24$.

Step3: Repeat the process

Divide the first term of the new dividend $-2x^{2}$ by the first term of the divisor $2x$, we get $-x$. Multiply $2x + 3$ by $-x$ to get $-2x^{2}-3x$. Subtract this from $-2x^{2}+0x - 24$: $(-2x^{2}+0x - 24)-(-2x^{2}-3x)=3x - 24$.

Step4: One more time

Divide the first term of the new dividend $3x$ by the first term of the divisor $2x$, we get $\frac{3}{2}$. Multiply $2x + 3$ by $\frac{3}{2}$ to get $3x+\frac{9}{2}$. Subtract this from $3x - 24$: $(3x - 24)-(3x+\frac{9}{2})=-\frac{57}{2}$.

Answer:

$2x^{2}-x+\frac{3}{2}-\frac{57}{2(2x + 3)}$