To find the area of the base of a cylinder given the volume and height, you can use the formula for the volume of a cylinder:
\[ \text{Volume} = \text{Base Area} \times \text{Height} \]
From the problem, we know the volume of the cylinder is represented by the polynomial:
\[ V = 2x^3 + 5x^2 + 3x + 18 \]
And the height of the cylinder is represented by the polynomial:
\[ h = x + 3 \]
To find the area of the base, we need to solve for the base area (B) using the formula:
\[ B = \frac{V}{h} \]
We will divide the polynomial representing the volume \( V \) by the polynomial representing the height \( h \):
\[ B = \frac{2x^3 + 5x^2 + 3x + 18}{x + 3} \]
Now, we will perform polynomial long division.
- Divide the first term of the numerator by the first term of the denominator:
\[ \frac{2x^3}{x} = 2x^2 \]
- Multiply \( 2x^2 \) by the entire denominator \( x + 3 \):
\[ 2x^2(x + 3) = 2x^3 + 6x^2 \]
- Subtract this result from the original polynomial:
\[ (2x^3 + 5x^2 + 3x + 18) - (2x^3 + 6x^2) = -x^2 + 3x + 18 \]
- Now, divide the first term of the new polynomial by the first term of the denominator again:
\[ \frac{-x^2}{x} = -x \]
- Multiply \(-x\) by the entire denominator:
\[ -x(x + 3) = -x^2 - 3x \]
- Subtract this from the result we got in the previous step:
\[ (-x^2 + 3x + 18) - (-x^2 - 3x) = 6x + 18 \]
- Now, divide the first term of the new polynomial \( 6x + 18 \) by the first term of the denominator:
\[ \frac{6x}{x} = 6 \]
- Multiply \( 6 \) by the entire denominator:
\[ 6(x + 3) = 6x + 18 \]
- Subtract this from \( 6x + 18 \):
\[ (6x + 18) - (6x + 18) = 0 \]
Thus, there is no remainder. The entire expression simplifies to:
\[ B = 2x^2 - x + 6 \]
So, the expression that represents the area of the base of the cylinder is:
\[ \boxed{2x^2 - x + 6} \]