To solve the given equation, we need to simplify the left-hand side and set it equal to the right-hand side. Let’s go through the steps one by one.
### Step-by-Step Solution
1. Combine Like Terms on the Left-Hand Side:
Given:
[tex]$
\left(3x^2 + 22x + 7\right) + (x + 7)
$[/tex]
Combine like terms:
- The constant terms: [tex]\(7 + 7 = 14\)[/tex]
- The linear terms: [tex]\(22x + x = 23x\)[/tex]
- The quadratic term remains as it is: [tex]\(3x^2\)[/tex]
Therefore, the left-hand side equation simplifies to:
[tex]$
3x^2 + 23x + 14
$[/tex]
2. Equate the Simplified Expression to the Right-Hand Side:
Now we have:
[tex]$
3x^2 + 23x + 14 = 3x + 1
$[/tex]
3. Interpret the Sentence to be Completed:
The question provides that:
[tex]$
(x + 7) \square = \square
$[/tex]
From the right-hand side of the equation:
- We need to express it in a form that supports a polynomial multiplication check.
Since we have [tex]\(3x^2 + 23x + 14\)[/tex] on the left-hand side and [tex]\(3x + 1\)[/tex] on the right, the natural expression for [tex]\((x + 7) \square\)[/tex] should be:
[tex]$
(x + 7)(3x + 2) = (3x + 2)
$[/tex]
So the complete sentence should be:
[tex]$
If \left(3 x^2+22 x+7\right)+(x+7)=3 x+1, \text{ then } (x+7) (3x + 1) = (3x + 1)
$[/tex]
On checking the polynomial division and multiplication, the product of two polynomials forming another polynomial ensures the property of polynomial multiplication holds. Thus, this sentence completion verifies the polynomial laws are satisfied.
### Final Answer:
Fill in the blanks as follows:
If [tex]\( \left(3 x^2+22 x+7\right)+(x+7)=3 x+1 \)[/tex], then [tex]\((x+7)\)[/tex] is multiplied by [tex]\( (3x + 1) = (3x + 1) \)[/tex].
The check of the polynomial division problem shows that the product of two polynomials is a polynomial. This supports the fact that the property of closure is satisfied for polynomial multiplication.
Done.
