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an architect is designing a new building. he makes a model of the building such that the area of the rectangular base is $12x^2 - 11x - 5$ and the length is $3x + 1$.
what is the width of the model in terms of $x$?

Explanation:

Step1: Recall the formula for the area of a rectangle

The area \( A \) of a rectangle is given by \( A = \text{length} \times \text{width} \). So, to find the width, we can use the formula \( \text{width} = \frac{\text{area}}{\text{length}} \).

Step2: Substitute the given area and length into the formula

We are given that the area is \( 12x^2 - 11x - 5 \) and the length is \( 3x + 1 \). So, we need to perform the division \( \frac{12x^2 - 11x - 5}{3x + 1} \). We can do this by factoring the numerator or by polynomial long division. Let's try factoring the numerator.

We need to factor \( 12x^2 - 11x - 5 \). We look for two numbers that multiply to \( 12 \times (-5) = -60 \) and add up to \( -11 \). The numbers are \( -15 \) and \( 4 \) because \( -15 \times 4 = -60 \) and \( -15 + 4 = -11 \).

So, we can rewrite the middle term:
\[
12x^2 - 15x + 4x - 5
\]

Now, factor by grouping:
\[
(12x^2 - 15x) + (4x - 5) = 3x(4x - 5) + 1(4x - 5) = (3x + 1)(4x - 5)
\]

Step3: Divide the factored area by the length

Now, we divide the factored area \( (3x + 1)(4x - 5) \) by the length \( 3x + 1 \):
\[
\frac{(3x + 1)(4x - 5)}{3x + 1}
\]

The \( 3x + 1 \) terms cancel out (assuming \( 3x + 1
eq 0 \)), leaving us with \( 4x - 5 \).

Answer:

\( 4x - 5 \)