3. A rectangular table has an area of (15x ^ 2 + 19x + 6) square units. Find the possible dimension of the table in terms of x.

We are tasked with finding the possible dimensions of a rectangular table whose area is given by the expression:

\[
A = 15x^2 + 19x + 6
\]

Since the area of a rectangle is the product of its length and width, we need to factor the quadratic expression \(15x^2 + 19x + 6\) into two binomials.

### Step-by-step factoring:

We are looking for two binomials of the form:
\[
A = (Ax + B)(Cx + D)
\]
such that:
\[
(15x^2 + 19x + 6) = (Ax + B)(Cx + D)
\]

First, consider the factors of \(15x^2\) and \(6\):

1. \(15x^2\) can be factored as \(15x \times x\), or \(3x \times 5x\).
2. \(6\) can be factored as \(1 \times 6\), \(2 \times 3\), or \(3 \times 2\).

Now, we will use trial and error to find the combination of factors that will give the middle term \(19x\).

### Factorization:

The correct factorization is:
\[
15x^2 + 19x + 6 = (3x + 2)(5x + 3)
\]

### Verification:

Let's expand \( (3x + 2)(5x + 3) \) to ensure the factorization is correct:
\[
(3x + 2)(5x + 3) = 3x(5x) + 3x(3) + 2(5x) + 2(3)
\]
\[
= 15x^2 + 9x + 10x + 6 = 15x^2 + 19x + 6
\]

The factorization is correct.

### Conclusion:

The possible dimensions of the rectangular table in terms of \(x\) are \(3x + 2\) and \(5x + 3\).

Would you like further details or clarification?

Here are 5 related questions to extend this concept:
1. How can you verify the factorization of a quadratic expression by expanding?
2. What are the possible dimensions if \(x = 2\)?
3. How do you determine the signs when factoring a trinomial?
4. What happens if the middle term had been negative?
5. Can all quadratics be factored into binomials?

**Tip:** Always expand your factored form to ensure it's correct.

A rectangular table has an area of (15x^2 + 19x + 6) square units. Find the possible dimension of the table in terms of x.

Explore how to determine the dimensions of a rectangular table with an area of 15x^2 + 19x + 6 square units. Learn to factor quadratic expressions to find the length and width in terms of x. Suitable for students in Grades 8-10.

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