To simplify the expression [tex]\((2x + 3y - 4)(-3x + y + 2)\)[/tex], we will use the distributive property (also known as the FOIL method for binomials). Each term in the first polynomial is multiplied by each term in the second polynomial.
Let's break this down step by step:
1. First, distribute [tex]\(2x\)[/tex]:
[tex]\[ 2x \cdot (-3x) = -6x^2 \][/tex]
[tex]\[ 2x \cdot y = 2xy \][/tex]
[tex]\[ 2x \cdot 2 = 4x \][/tex]
2. Next, distribute [tex]\(3y\)[/tex]:
[tex]\[ 3y \cdot (-3x) = -9xy \][/tex]
[tex]\[ 3y \cdot y = 3y^2 \][/tex]
[tex]\[ 3y \cdot 2 = 6y \][/tex]
3. Finally, distribute [tex]\(-4\)[/tex]:
[tex]\[ -4 \cdot (-3x) = 12x \][/tex]
[tex]\[ -4 \cdot y = -4y \][/tex]
[tex]\[ -4 \cdot 2 = -8 \][/tex]
Now, let's combine all these terms:
[tex]\[ -6x^2 + 2xy + 4x - 9xy + 3y^2 + 6y + 12x - 4y - 8 \][/tex]
Next, we combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[ -6x^2 \][/tex]
- Combine the [tex]\(xy\)[/tex] terms:
[tex]\[ 2xy - 9xy = -7xy \][/tex]
- Combine the [tex]\(x\)[/tex] terms:
[tex]\[ 4x + 12x = 16x \][/tex]
- Combine the [tex]\(y\)[/tex] terms:
[tex]\[ 6y - 4y = 2y \][/tex]
- The [tex]\(y^2\)[/tex] term remains:
[tex]\[ 3y^2 \][/tex]
- The constant term remains:
[tex]\[ -8 \][/tex]
Bringing it all together, we get:
[tex]\[ -6x^2 + 16x - 7xy + 3y^2 + 2y - 8 \][/tex]
So, the simplified expression is:
[tex]\[ -6x^2 + 16x - 7xy + 3y^2 + 2y - 8 \][/tex]
Looking at the provided options, the correct answer is:
[tex]\[ -6x^2 + 16x - 7xy + 3y^2 + 2y - 8 \][/tex]
