Question

Fill in the blanks to correctly subtract the polynomials.
(m^2 + 6m + [?]) - (2m^2 - [?]m +1) = -m^2 + 9m + 4

Answer 1

(3x^2 + 2x − 1) + (2x^2 − 4x + 3)

Answer 2

To solve the polynomial subtraction \((m^2 + 6m + [?]) - (2m^2 - [?]m + 1) = -m^2 + 9m + 4\), we need to fill in the blanks such that the equation holds true.

Let's denote \( a = ? \) and \( b = ? \) for convenience:

1. Start with the left-hand side of the equation:
\[
(m^2 + 6m + a) - (2m^2 - bm + 1)
\]
This simplifies to:
\[
m^2 + 6m + a - 2m^2 + bm - 1 = (-m^2 + (6 + b)m + (a - 1))
\]

2. Now, we want this expression to equal \(-m^2 + 9m + 4\):
\[
-m^2 + (6 + b)m + (a - 1) = -m^2 + 9m + 4
\]

3. From this comparison, we can equate the coefficients:

- For the \(m\) term:
\[
6 + b = 9
\]
Solving for \(b\), we subtract 6 from both sides:
\[
b = 9 - 6 = 3
\]

- For the constant term:
\[
a - 1 = 4
\]
Solving for \(a\), we add 1 to both sides:
\[
a = 4 + 1 = 5
\]

Now, we can fill in the blanks:
\[
(m^2 + 6m + [5]) - (2m^2 - [3]m + 1) = -m^2 + 9m + 4
\]

Thus, the values to fill in the blanks are:
- The first blank: **5**
- The second blank: **3**

Final answer:
\[
\text{Fill in the blanks with "5" and "3".}
\]

Answer 3

To add the polynomials \((3x^2 + 2x - 1) + (2x^2 - 4x + 3)\), we combine like terms.

1. **Combine the \(x^2\) terms:**
\[
3x^2 + 2x^2 = 5x^2
\]

2. **Combine the \(x\) terms:**
\[
2x - 4x = -2x
\]

3. **Combine the constant terms:**
\[
-1 + 3 = 2
\]

Now, putting it all together, we have:
\[
5x^2 - 2x + 2
\]

So, the result of the addition is:
\[
\boxed{5x^2 - 2x + 2}
\]