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.

To find the possible dimensions of the rectangular table, we need to factor the given expression for the area, which is \( 15x^2 + 19x + 6 \). The area of a rectangle is the product of its length and width, so factoring this quadratic expression will give us the dimensions in terms of \( x \).

### Step 1: Factor the quadratic expression
We need to factor the expression \( 15x^2 + 19x + 6 \). We will use the trial-and-error method (also known as factoring by grouping) to find the factors.

1. Multiply the coefficient of \( x^2 \) (which is 15) by the constant term (which is 6). This gives us \( 15 \times 6 = 90 \).
2. We need to find two numbers that multiply to 90 and add to 19 (the middle term coefficient).

   The pair of numbers that satisfy this are 10 and 9, because:
   \[
   10 \times 9 = 90 \quad \text{and} \quad 10 + 9 = 19
   \]

### Step 2: Rewrite the middle term
Now we rewrite the quadratic as:
\[
15x^2 + 10x + 9x + 6
\]

### Step 3: Factor by grouping
Group the terms:
\[
(15x^2 + 10x) + (9x + 6)
\]

Now factor out the greatest common factor (GCF) from each group:
\[
5x(3x + 2) + 3(3x + 2)
\]

### Step 4: Factor out the common binomial factor
Since both groups have the factor \( (3x + 2) \), we can factor it out:
\[
(5x + 3)(3x + 2)
\]

### Conclusion:
The factored form of \( 15x^2 + 19x + 6 \) is \( (5x + 3)(3x + 2) \). Therefore, the possible dimensions of the rectangular table are \( (5x + 3) \) units and \( (3x + 2) \) units.

Would you like further details or have any questions?

Here are five related questions:
1. How do you check if a factorization is correct?
2. What if one of the dimensions was provided? How would you find the other?
3. Can this problem be solved using the quadratic formula instead of factoring?
4. How would you factor more complex quadratic expressions?
5. How can you determine the GCF in a polynomial with higher powers?

**Tip:** When factoring quadratics, always double-check your work by expanding the factors to verify the original expression!

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

Discover how to find the dimensions of a rectangular table with an area of (15x^2 + 19x + 6) square units. This problem involves factoring quadratic expressions and understanding the relationship between area, length, and width. Suitable for students in Grades 8-10.

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