a)
$P(x) = 3x^{4} + 5x³ + x² - 2x² - \frac{1}{4}x - 2 = 3x^{4} + 5x³ - x² - \frac{1}{4}x - 2$
$Q(x) = 3x^{4} - 3x³ + x² - x² - \frac{1}{4} = 3x^{4} - 3x³ - \frac{1}{4}$
b)
$P(x) + Q(x)$
$= 3x^{4} + 5x³ - x² - \frac{1}{4}x - 2 + 3x^{4} - 3x³ - \frac{1}{4}$
$= (3 + 3)x^{4} + (5 - 3)x³ - x²- \frac{1}{4}x - 2 - \frac{1}{4}$
$= 6x^{4} + 2x³ - x² - \frac{1}{4}x - \frac{9}{4}$
$P(x) - Q(x)$
$= 3x^{4} + 5x³ - x² - \frac{1}{4}x - 2- 3x^{4} + 3x³ + \frac{1}{4}$
$ = (3 - 3)x^{4} + (5 + 3)x³ - x²- \frac{1}{4}x - 2 + \frac{1}{4}$
$= 8x³ - x² - \frac{1}{4}x - \frac{7}{4}$
c)
$2P(x) + 5Q(x)$
$= 2(3x^{4} + 5x³ - x² - \frac{1}{4}x - 2) + 5(3x^{4} - 3x³ - \frac{1}{4})$
$= 6x^{4} + 10x³ - 2x² - \frac{1}{2}x - 4 + 15x^{4} - 15x³ - \frac{5}{4}$
$= (6 + 15)x^{4} + (10 - 15)x³ - 2x²- \frac{1}{2}x - 4 - \frac{5}{4}$
$= 21x^{4} - 5x³ - 2x² - \frac{1}{2}x - \frac{21}{4}$
$4P(x) - 3Q(x)$
$= 4(3x^{4} + 5x³ - x² - \frac{1}{4}x - 2) - 3(3x^{4} - 3x³ - \frac{1}{4})$
$= 12x^{4} + 20x³ - 4x² - x - 8 - 9x^{4} + 9x³ + \frac{3}{4}$
$= (12 - 9)x^{4} + (20 + 9)x³ - 4x²- x - 8 + \frac{3}{4}$
$= 3x^{4} + 29x³ - 4x² - x - \frac{29}{4}$
