exam Maths Puzzle Numerical Ability

can someone explain to me why are polynomials important?

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Help please:

Divide--

p^2 - q^2/p+q divided by p-q/p+q subtract the following:

x^2+2x/3x - x^2+5x+6/3x

I came up with x^4+5x+6/3x...is that right?

I need step by step if my answer is wrong. Thank you so much for all your help.