Fill in the blanks to correctly subtract the polynomials.

Question

(m^2 + 6m + [?]) - (2m^2 - [?]m +1) = -m^2 + 9m + 4

Answer 1

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 2

(3x^2 + 2x − 1) + (2x^2 − 4x + 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: \[ \boxed{5x^2 - 2x + 2} \]