Multiplying Algebraic Expressions

To multiply (2x + 3) and (x - 4), we use the distributive property. First, we multiply 2x by x and 2x by -4, which gives us 2x^2 and -8x. Then, we multiply 3 by x and 3 by -4, which gives us 3x and -12. Finally, we add the like terms to get 2x^2 - 5x - 12.

To multiply (x^2 - 2x + 1) and (x + 3), we use the distributive property. First, we multiply x^2 by x and x^2 by 3, which gives us x^3 and 3x^2. Then, we multiply -2x by x and -2x by 3, which gives us -2x^2 and -6x. Finally, we multiply 1 by x and 1 by 3, which gives us x and 3. We then add the like terms to get x^3 + x^2 - 5x + 3.

To multiply (3x^2y - 2xy + 4) and (2x - 3y), we use the distributive property. First, we multiply 3x^2y by 2x and 3x^2y by -3y, which gives us 6x^3y^2 and -9x^2y^2. Then, we multiply -2xy by 2x and -2xy by -3y, which gives us -4x^2y and 6xy. Finally, we multiply 4 by 2x and 4 by -3y, which gives us 8x and -12y. We then add the like terms to get 6x^3y^2 - 10x^2y^2 + 11xy - 12y.

To multiply (x + 2)(x - 2)(x^2 + 4), we first multiply (x + 2) and (x - 2) using the difference of squares formula. This gives us x^2 - 4. Then, we multiply x^2 - 4 by x^2 + 4 using the sum of cubes formula. This gives us x^4 - 16.

To multiply (x^3 - 2x^2 + x - 2)(x^2 + 3x - 4), we use the distributive property. First, we multiply x^3 by x^2, x^3 by 3x, x^3 by -4, -2x^2 by x^2, -2x^2 by 3x, -2x^2 by -4, x by x^2, x by 3x, x by -4, -2 by x^2, -2 by 3x, and -2 by -4. This gives us x^5, 3x^4, -4x^3, -2x^4, -6x^3, 8x^2, x^2, 3x, -4, -2x^2, -6x, and 8. We then add the like terms to get x^5 + x^4 - 7x^3 - 14x^2 + 11x + 8.

To multiply (2x + 3y)(3x - 2y), we use the distributive property. First, we multiply 2x by 3x and 2x by -2y, which gives us 6x^2 and -4xy. Then, we multiply 3y by 3x and 3y by -2y, which gives us 9xy and -6y^2. We then add the like terms to get 6x^2 - 9xy + 6y^2.

To multiply (x^2 + 2x + 1)(x - 1), we use the distributive property. First, we multiply x^2 by x and x^2 by -1, which gives us x^3 and -x^2. Then, we multiply 2x by x and 2x by -1, which gives us 2x^2 and -2x. Finally, we multiply 1 by x and 1 by -1, which gives us x and -1. We then add the like terms to get x^3 + x^2 - 1.

To multiply (x^3 - 2x^2 + x - 2)(x + 1), we use the distributive property. First, we multiply x^3 by x and x^3 by 1, which gives us x^4 and x^3. Then, we multiply -2x^2 by x and -2x^2 by 1, which gives us -2x^3 and -2x^2. Next, we multiply x by x and x by 1, which gives us x^2 and x. Finally, we multiply -2 by x and -2 by 1, which gives us -2x and -2. We then add the like terms to get x^4 - x^3 - x^2 + x - 2.

To multiply (2x^2 - 3x + 4)(3x^2 + 2x - 1), we use the distributive property. First, we multiply 2x^2 by 3x^2, 2x^2 by 2x, 2x^2 by -1, -3x by 3x^2, -3x by 2x, -3x by -1, 4 by 3x^2, 4 by 2x, and 4 by -1. This gives us 6x^4, 4x^3, -2x^2, -9x^3, -6x^2, 3x, 12x^2, 8x, and -4. We then add the like terms to get 6x^4 - 5x^3 - 11x^2 + 11x - 4.

To multiply (x^2 - 2x + 3)(x^2 + 2x - 3), we use the difference of squares formula. This gives us (x^2 - 2x + 3)(x^2 + 2x - 3) = (x^2)^2 - (2x - 3)^2 = x^4 - (4x^2 - 12x + 9) = x^4 - 4x^2 + 12x - 9.

To multiply (x^3 + 2x^2 - 3x + 4)(x^3 - 2x^2 + 3x - 4), we use the difference of squares formula. This gives us (x^3 + 2x^2 - 3x + 4)(x^3 - 2x^2 + 3x - 4) = (x^3)^2 - (2x^2 - 3x + 4)^2 = x^6 - (4x^4 - 12x^3 + 25x^2 - 24x + 16) = x^6 - 4x^4 + 12x^3 - 25x^2 + 24x - 16.

To multiply (2x^2 + 3x - 4)(2x^2 - 3x + 4), we use the difference of squares formula. This gives us (2x^2 + 3x - 4)(2x^2 - 3x + 4) = (2x^2)^2 - (3x - 4)^2 = 4x^4 - (9x^2 - 24x + 16) = 4x^4 - 9x^2 + 24x - 16.

To multiply (x^4 + 2x^3 - 3x^2 + 4x - 5)(x^4 - 2x^3 + 3x^2 - 4x + 5), we use the difference of squares formula. This gives us (x^4 + 2x^3 - 3x^2 + 4x - 5)(x^4 - 2x^3 + 3x^2 - 4x + 5) = (x^4)^2 - (2x^3 - 3x^2 + 4x - 5)^2 = x^8 - (4x^6 - 12x^5 + 25x^4 - 40x^3 + 60x^2 - 40x + 25) = x^8 - 4x^6 + 12x^5 - 25x^4 + 40x^3 - 60x^2 + 40x - 25.

To multiply (x^5 + 2x^4 - 3x^3 + 4x^2 - 5x + 6)(x^5 - 2x^4 + 3x^3 - 4x^2 + 5x - 6), we use the difference of squares formula. This gives us (x^5 + 2x^4 - 3x^3 + 4x^2 - 5x + 6)(x^5 - 2x^4 + 3x^3 - 4x^2 + 5x - 6) = (x^5)^2 - (2x^4 - 3x^3 + 4x^2 - 5x + 6)^2 = x^10 - (4x^8 - 12x^7 + 25x^6 - 40x^5 + 60x^4 - 40x^3 + 25x^2 - 60x + 36) = x^10 - 4x^8 + 12x^7 - 25x^6 + 40x^5 - 60x^4 + 40x^3 - 25x^2 + 60x - 36.