Questions: Perform the indicated operations. (2 x^2-2 x y-3 y^2)-(x^2+2 x y-4 y^2)+(3 x y+y^2) (Simplify your answer. Type your answer in standard form.)

Transcript text: Perform the indicated operations. \[ \left(2 x^{2}-2 x y-3 y^{2}\right)-\left(x^{2}+2 x y-4 y^{2}\right)+\left(3 x y+y^{2}\right) \] (Simplify your answer. Type your answer in standard form.)

Solution

Solution Steps

To simplify the given expression, we need to combine like terms. This involves distributing any negative signs through the parentheses and then adding or subtracting the coefficients of the same terms.

Step 1: Expand the Expression

We start with the expression: \[ \left(2 x^{2}-2 x y-3 y^{2}\right)-\left(x^{2}+2 x y-4 y^{2}\right)+\left(3 x y+y^{2}\right) \] We will distribute the negative sign through the second term and then combine all terms.

Step 2: Combine Like Terms

After distributing the negative sign, the expression becomes: \[ 2x^{2} - 2xy - 3y^{2} - x^{2} - 2xy + 4y^{2} + 3xy + y^{2} \] Now, we combine the like terms:

For \(x^{2}\): \(2x^{2} - x^{2} = x^{2}\)
For \(xy\): \(-2xy - 2xy + 3xy = -xy\)
For \(y^{2}\): \(-3y^{2} + 4y^{2} + y^{2} = 2y^{2}\)

Step 3: Write the Simplified Expression

Combining all the results, we obtain the simplified expression: \[ x^{2} - xy + 2y^{2} \]