To solve the problem, we need to find expressions that combine to equal \(3x^2 + 4x + 5\).
Let's analyze the options provided:
- \(2x^2 + 2x + 2\)
- ?
- ?
- ?
- \(x^2 + 6x - 1\)
- ?
- \(-x - 1\)
Now, let’s label the options a, b, c, and d to investigate which can be added together with the given expressions to yield \(3x^2 + 4x + 5\).
We'll calculate the expressions while keeping the target expression in mind:
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Analyzing the sum of \(2x^2 + 2x + 2\):
- Let's say we take option a, \( -x + 3 \):
- \(2x^2 + 2x + 2 + (-x + 3) = 2x^2 + (2x - x) + (2 + 3) = 2x^2 + x + 5\)
This does not yield \(3x^2 + 4x + 5\).
- Let's say we take option a, \( -x + 3 \):
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Let's instead use \(b\), \(x^2 + 3x + 4\):
- Adding: \[2x^2 + 2x + 2 + (x^2 + 3x + 4) = (2x^2 + x^2) + (2x + 3x) + (2 + 4) = 3x^2 + 5x + 6\]
This one also does not yield the desired expression!
-
Let's try \(c\), \(-3x + 1\):
- Now we add it: \[ 2x^2 + 2x + 2 + (-3x + 1) = 2x^2 + (2x - 3x) + (2 + 1) = 2x^2 - x + 3 \] Not correct.
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What if we take the last option \(d\), \(2x^2 - x + 7\):
- Adding \(2x^2 + 2x + 2\) to this: \[ 2x^2 + 2x + 2 + (2x^2 - x + 7) = (2x^2 + 2x^2) + (2x - x) + (2 + 7) = 4x^2 + x + 9 \]
Given this, we should see that our method may not work simply by checking them in sequence. It’s better to systematically construct the expression \(3x^2 + 4x + 5\) by finding others needed to complete the adding step.
Here is a valid approach using them strategically:
We can isolate combinations:
Use:
- \(2x^2 + 2x + 2\)
- Try \(b\) or \(c\) followed by others till sequence aligns properly.
But after these bases, I will follow through backtracking to simplify via checking again thoroughly.
Finally identifying piecemail, you will find out:
- 2x^2 + 2x + 2
- -3x + 1
- x^2 + 6x - 1
- -x - 1
Yields complete sequence.
The answers can be filled in order:
- 2x^2 + 2x + 2
- -3x + 1 (c)
- x^2 + 6x - 1
- -x - 1
- Following others appropriately till reaching total assembly.