a) M\(\left(x\right)=x^2+5x^4-3x^3+4x^2+x^4+3x^3-x+5\)
\(=\left(x^2+4x^2\right)+\left(5x^4+x^4\right)+\left(-3x^3+3x^3\right)-x+5\)
\(=5x^2+6x^4-x+5\)
\(=6x^4+5x^2-x+5\)
\(N\left(x\right)=x-5x^3-2x^2-8x^4+4x^3-x+5\)
\(=\left(x-x\right)+\left(5x^3+4x^3\right)-2x^2-8x^4+5\)
\(=0+9x^3-2x^2-8x^4+5\)
\(=\left(-8x^4\right)+9x^3-2x^2+5\)
b) M\(\left(x\right)+N\left(x\right)=\) \((x^2+5x^4-3x^3+4x^2+x^4+3x^3-x+5)\) \(+\)\((x-5x^3-2x^2-8x^4+4x^3-x+5)\)
\(=\) \(x^2+5x^4-3x^3+4x^2+x^4+3x^3-x+5\) \(+\)\(x-5x^3-2x^2-8x^4+4x^3-x+5\)
\(=\)\(\left(x^2+4x^2-2x^2\right)+\left(5x^4+x^4-8x^4\right)+\left(-3x^3+3x^3-5x^3+4x^3\right)\)\(+\left(-x+x-x\right)+\left(5+5\right)\)
\(=3x^2-2x^4+x^3-x+10\)
c)\(M\left(x\right)-N\left(x\right)\) \(=\) \((x^2+5x^4-3x^3+4x^2+x^4+3x^3-x+5)\) \(-\)\((x-5x^3-2x^2-8x^4+4x^3-x+5)\)
\(=\) \(x^2+5x^4-3x^3+4x^2+x^4+3x^3-x+5\) \(-x+5x^3+2x^2+8x^4-4x^3+x-5\)
\(=\) \(\left(x^2+4x^2+2x^2\right)+\left(5x^4+x^4+8x^4\right)+\left(-3x^3+3x^3+5x^3-4x^3\right)\) \(+\) \(\left(-x-x+x\right)+\left(5-5\right)\)
\(=7x^2+14x^4+x^3-x\)
