A rectangular playground has an area represented by 3x^2+14x+8. If its width is represented by 3x+2, what could be its length?

I just need to figure out how to solve it, not the actual answer.

Thanks!

since area = length*width,

length = area/width

length = (3x^2+14x+8)/(3x+2)

now just crank it out

To determine the length of the rectangular playground, we need to use the formula for the area of a rectangle, which is length multiplied by width.

In this case, the given area is represented by the quadratic expression 3x^2 + 14x + 8, and the given width is represented by the binomial expression 3x + 2.

To find the length, we need to divide the area by the width. This can be done by performing polynomial long division or by factoring out the width from the area expression.

Let's go through the steps of polynomial long division to find the quotient, which will represent the length:

1. First, set up the division in the following format:
_________________________
3x + 2 | 3x^2 + 14x + 8

2. Divide the first term of the dividend (3x^2) by the first term of the divisor (3x). The result is x. Write this above the division line.

x
_________________________
3x + 2 | 3x^2 + 14x + 8

3. Multiply the divisor (3x + 2) by the quotient (x) and write the product below the dividend, aligning like terms:

x
_________________________
3x + 2 | 3x^2 + 14x + 8
-(3x^2 + 2x)

4. Subtract the product you just found from the dividend:

x
_________________________
3x + 2 | 3x^2 + 14x + 8
-(3x^2 + 2x)
_______________
12x + 8

5. Divide the current term of the dividend (12x) by the first term of the divisor (3x) to get 4. Write this above the division line:

x + 4
_________________________
3x + 2 | 3x^2 + 14x + 8
-(3x^2 + 2x)
_______________
12x + 8
-(12x + 8)

6. Subtract the product of the divisor and quotient from the current dividend:

x + 4
_________________________
3x + 2 | 3x^2 + 14x + 8
-(3x^2 + 2x)
_______________
12x + 8
-(12x + 8)
_______________
0

Now, we have reached a remainder of 0, indicating that the division is exact. The quotient x + 4 represents the length of the rectangular playground.

Therefore, the possible lengths could be expressed as (3x + 2)(x + 4).