[tex]E(x) = {x}^{3} + {x}^{2} + x + 1[/tex]
[tex]a)E(0) = E( \sqrt{2} ) - 3 \sqrt{2} - 2[/tex]
[tex]E(0) = {0}^{3} + {0}^{2} + 0 + 1 = 1[/tex]
[tex]E( \sqrt{2} ) = { (\sqrt{2} )}^{3} + {( \sqrt{2}) }^{2} + \sqrt{2} + 1[/tex]
[tex]E( \sqrt{2} ) = 2 \sqrt{2} + 2 + \sqrt{2} + 1[/tex]
[tex]E( \sqrt{2} ) = 3 \sqrt{2} + 3[/tex]
[tex]1= 3 \sqrt{2} + 3 - 3 \sqrt{2} - 2[/tex]
[tex]1 = 3 - 2[/tex]
[tex]1 = 1 \: (A)[/tex]
[tex]b)2E(x) - {x}^{3} + 6 {x}^{2} - 9x - 11[/tex]
[tex] = 2( {x}^{3} + {x}^{2} + x + 1) - {x}^{3} + 6 {x}^{2} - 9x - 11[/tex]
[tex] = 2 {x}^{3} + 2 {x}^{2} + 2x + 2 - {x}^{3} + 6 {x}^{2} - 9x - 11[/tex]
[tex] = {x}^{3} + 8 {x}^{2} - 7x - 9[/tex]
[tex]c)E(x) - {x}^{3} + x \geqslant 0[/tex]
[tex] {x}^{3} + {x}^{2} + x + 1 - {x}^{3} + x \geqslant 0[/tex]
[tex] {x}^{2} + 2x + 1 \geqslant 0[/tex]
[tex] {(x + 1)}^{2} \geqslant 0 \: ,\forall \: \:x\:\in\:\mathbb{R}[/tex]
[tex]d)E(x) = {x}^{3} + {x}^{2} + x + 1[/tex]
[tex]E(x) = {x}^{2} (x + 1) + x + 1[/tex]
[tex]E(x) = (x + 1)( {x}^{2} + 1)[/tex]
