a)
[tex]( {x}^{2} + x - 1)( {x}^{2} - x + 1) = {x}^{4} - {x}^{3} + {x}^{2} + {x}^{3} - {x}^{2} + x - {x}^{2} + x - 1 = {x}^{4} - {x}^{2} + 2x - 1[/tex]
b)
[tex](2 {x}^{2} - 3x + 2)( {x}^{2} + 4x - 3) = 2 {x}^{4} + 8 {x}^{3} - 6 {x}^{2} - 3 {x}^{3} - 12 {x}^{2} + 9x + 2 {x}^{2} + 8x - 6 = 2 {x}^{4} + 5 {x}^{3} - 16 {x}^{2} + 17x - 6[/tex]
c)
[tex](2 {x}^{2} + \frac{1}{2} x + 1)( {x}^{2} - x - \frac{1}{4}) = 2 {x}^{4} - 2 {x}^{3} - \frac{1}{2} {x}^{2} + \frac{1}{2} {x}^{3} - \frac{1}{2} {x}^{2} - \frac{1}{8}x + {x}^{2} - x - \frac{1}{4} = 2 {x}^{4} - \frac{3}{2} {x}^{3} - \frac{9}{8} x - \frac{1}{4} [/tex]
d)
[tex](x - 1)(x + 2)(x - 3) = ( {x}^{2} + 2x - x - 2)(x - 3) = ( {x}^{2} + x - 2)(x - 3 )= {x}^{3} - 3 {x}^{2} + {x}^{2} - 3x - 2x + 6 = {x}^{3} - 2 {x}^{2} - 5x + 6 [/tex]
e)
[tex](2x - 1)(x + 3)(2x - 1) = (2 {x}^{2} + 6x - x - 3)(2x - 1) = (2 {x}^{2} + 5x - 3)(2x - 1) = 4 {x}^{3} - 2 {x}^{2} + 10 {x}^{2} - 5x - 6x + 3 = 4 {x}^{3} + 8 {x}^{2} - 11x + 3[/tex]
f)
[tex](x + \frac{1}{2})(2x + 4)(x + 4) = (2 {x}^{2} + 4x + x + 2)(x + 4) = (2 {x}^{2} + 5x + 2)(x + 4) = 2 {x}^{3} + 8 {x}^{2} + 5 {x}^{2} + 20x + 2x + 8 = 2 {x}^{3} + 13 {x}^{2} + 22x + 8[/tex]
