The area of a rectangular swimming pool is given by 4x^2+19x+12ft^2.One side length of the pool is given by 4x + 3 feet. What is an algebraic expression for the other side length of the pool?

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To find the algebraic expression for the other side length of the pool, we need to divide the area of the pool by the given side length and simplify the expression.

The area of the rectangular swimming pool is given by 4x^2 + 19x + 12 ft^2, and one side length is given by 4x + 3 feet.

Let's divide the area by the given side length:

(4x^2 + 19x + 12 ft^2) / (4x + 3 ft)

To divide this expression, we can use polynomial division. The first step is to divide the leading terms:

(4x^2) / (4x) = x

So, the quotient so far is x.

Now, we need to multiply this quotient by the divisor (4x + 3 ft):

x * (4x + 3 ft) = 4x^2 + 3x ft

Next, we subtract this product from the original polynomial:

(4x^2 + 19x + 12 ft^2) - (4x^2 + 3x ft) = (16x + 12 ft^2)

Now, we have a new polynomial (16x + 12 ft^2). Let's divide the leading term again:

(16x) / (4x) = 4

The quotient now becomes x + 4.

Multiplying this quotient (x + 4) by the divisor (4x + 3 ft):

(x + 4) * (4x + 3 ft) = 4x^2 + 13x + 12 ft^2

Finally, subtracting this product from the new polynomial:

(16x + 12 ft^2) - (4x^2 + 13x + 12 ft^2) = 3x

Therefore, the algebraic expression for the other side length of the pool is 3x feet.