Alan

|x+1| = x - 1   has no solution.

n(n + 3) - 4  =  n^2 + 3n - 4 = (n + 4)(n - 1)

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A single penalty kick is just one player taking a penalty against one goalkeeper.

There are 19 players who take a penalty kick against goalkeeper number one (that's 19 penalty kicks).

There are 19 players who take a penalty kick against goalkeeper number two (that's another 19 penalty kicks).

There are 19 players who take a penalty kick against goalkeeper number three (that's yet another 19 penalty kicks).

So how many in total?

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I agree with Daisy's result.  

In the following let u = up, d = down and x - either up or down.

The only possible ways in which Polya stays on floor 7 or above at every stop are as follows (where moves are ordered from left to right):

u u u x x   p1 = (1/2)³

u u d u u   p2 = (1/2)⁵

u u d u d   p3 = (1/2)⁵

p1+p2+p3 = 3/16

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Look at Melody's answer here: https://web2.0calc.com/questions/help_94831#r3 

"A function f has a horizontal asymptote of y = -4 a vertical asymptote of x = 3 and an x-intercept at (1,0). Part (a): Let f be of the form f(x) = \frac{ax+b}{x+c}.Find an expression for f(x). "

\(f(x)=\frac{ax+b}{x+c}\)

Horizontal asymptote means set x to infinity so \(f(x)\rightarrow a\)  hence we immediately have \(a=-4\)

Vertical asymptote means set denominator to zero at x = 3, so \(3+c=0 \text{ or } c = -3\)  

x intercept at (1, 0) means \(\frac{a\times1+b}{1+c}=0\)  so  \(\frac{-4+b}{1-3}=0 \text{ or }b = 4\)  

Hence \(f(x)=\frac{-4x+4}{x-3}\text{ or }f(x)=\frac{4(1-x)}{x-3}\) 

Use the same approach for part b.

(Edited to correct silly mistake!  Thanks heureka.)

Thanks. It’s a method I’ve only seen used on cubic equations before - to eliminate the x² term. I guess the quartic specified here was deliberately chosen to simplify enough that it would be easy to find the roots of the resulting transformed function.

I'm not entirely sure the following is exactly the Tschirnhaus transformation method, but it's as close as I can get!

The points would lie on one of two lines: y = 2x and y = x/2

These split the plane into four regions.

Yes.  I'm afraid I gave the positive roots of the equation the labels m and n, but, had I looked more carefully, I would have noticed that your definition of m and n referred to the values under the square root sign!