3x ⁴ = 3x ² • x ². Then
(3x ⁴ - 2x ³ + 4x - 5) - 3x ² (x ² + 4) = -2x ³ - 12x ² + 4x - 5
-2x ³ = -2x • x ². Then
(-2x ³ - 12x ² + 4x - 5) - (-2x) (x ² + 4) = -12x ² + 12x - 5
-12x ² = -12 • x ². Then
(-12x ² + 12x - 5) - (-12) (x ² + 4) = 12x + 43
So we've shown
[tex]\displaystyle \frac{3x^4-2x^3+4x-5}{x^2+4} = 3x^2 - \frac{2x^3+12x^2-4x+5}{x^2+4} \\\\ = 3x^2 - 2x - \frac{12x^2-12x+5}{x^2+4} \\\\ = \boxed{3x^2 - 2x - 7 + \frac{12x+43}{x^2+4}}[/tex]
