$\text{a)}$  

$w\left(x\right)+u\left(x\right)=2\sqrt{3}{x}^4-5x+1+x^4-3x^2+4x$  

$w\left(x\right)+u\left(x\right)=\left(2\sqrt{3}+1\right)x^4-3x^2-x+1$  

Zatem otrzymujemy:

$a_4=2\sqrt{3}+1$  

$a_3=0$  

$a_2=-3$  

$a_1=-1$  

$a_0=1$  

 

$2w\left(x\right)-u\left(x\right)=2\cdot \left(2\sqrt{3}{x}^4-5x+1\right)-\left(x^4-3x^2+4x\right)$  

$2w\left(x\right)-u\left(x\right)=4\sqrt{3}{x}^4-10x+2-x^4+3x^2-4x$  

$2w\left(x\right)-u\left(x\right)=\left(4\sqrt{3}-1\right)x^4+3x^2-14x+2$  

Zatem otrzymujemy:

$a_4=4\sqrt{3}-1$  

$a_3=0$  

$a_2=3$  

$a_1=-14$  

$a_0=2$  

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$\text{b)}$  

$w\left(x\right)+u\left(x\right)=-x^3+\sqrt{2}{x}^2+3\sqrt{2}+6x^3-3\sqrt{2}{x}^2-1$  

$w\left(x\right)+u\left(x\right)=5x^3-2\sqrt{2}{x}^2+3\sqrt{2}-1$  

Zatem otrzymujemy:

$a_3=5$  

$a_2=-2\sqrt{2}$  

$a_1=0$  

$a_0=3\sqrt{2}-1$  

 

$2w\left(x\right)-u\left(x\right)=2\cdot \left(-x^3+\sqrt{2}{x}^2+3\sqrt{2}\right)-\left(6x^3-3\sqrt{2}{x}^2-1\right)$  

$2w\left(x\right)-u\left(x\right)=-2x^3+2\sqrt{2}{x}^2+6\sqrt{2}-6x^3+3\sqrt{2}{x}^2+1$  

$2w\left(x\right)-u\left(x\right)=-8x^3+5\sqrt{2}{x}^2+6\sqrt{2}+1$  

Zatem otrzymujemy:

$a_3=-8$  

$a_2=5\sqrt{2}$  

$a_1=0$  

$a_0=6\sqrt{2}+1$