Let's tackle the problem of factoring the quadratic expression [tex]\(2x^2 + 7x + 6\)[/tex]. To factorize this, we need to express it as a product of two binomials.
### Step-by-Step Solution:
1. Identify the quadratic expression:
We start with the quadratic expression [tex]\(2x^2 + 7x + 6\)[/tex].
2. Factor the expression:
We want to factor the quadratic expression [tex]\(2x^2 + 7x + 6\)[/tex] into the form [tex]\((ax + b)(cx + d)\)[/tex].
Through factoring, we find that:
[tex]\[
2x^2 + 7x + 6 = (x + 2)(2x + 3)
\][/tex]
3. Equation of Area Representation:
To represent this algebraically, we express that the area of the rectangle formed by sides [tex]\((x + 2)\)[/tex] and [tex]\((2x + 3)\)[/tex] is equivalent to the original quadratic expression.
Therefore:
[tex]\[
\text{Area} = \text{Length} \times \text{Width}
\][/tex]
[tex]\[
2x^2 + 7x + 6 = (x + 2)(2x + 3)
\][/tex]
4. Verify by Multiplying the Factors to get the Original Expression:
Let's expand the factors to ensure we get back the original expression:
[tex]\[
(x + 2)(2x + 3) = x \cdot 2x + x \cdot 3 + 2 \cdot 2x + 2 \cdot 3
\][/tex]
[tex]\[
= 2x^2 + 3x + 4x + 6
\][/tex]
[tex]\[
= 2x^2 + 7x + 6
\][/tex]
Thus, the factors multiply to give us the original quadratic expression, confirming that our factorization is correct.
### Rectangular Sketch:
To visualize it, you can imagine the rectangle with sides [tex]\(x + 2\)[/tex] and [tex]\(2x + 3\)[/tex]:
- The length of the rectangle is [tex]\(x + 2\)[/tex].
- The width of the rectangle is [tex]\(2x + 3\)[/tex].
### Conclusion:
We have successfully factored the quadratic expression [tex]\(2x^2 + 7x + 6\)[/tex] into [tex]\((x + 2)(2x + 3)\)[/tex]. This means it can be represented as a rectangular area where one side is [tex]\(x + 2\)[/tex] and the other side is [tex]\(2x + 3\)[/tex]. The area of this rectangle matches the original quadratic expression when multiplied out, confirming the factorization.
Thus, [tex]\(2x^2 + 7x + 6\)[/tex] can indeed be factored, and the area equivalently represents the product of its side length factors.
