Alan

I prefer to draw a graph first, for these sort of questions:

Find a monic quartic polynomial f(x) with rational coefficients whose roots include x=1-\sqrt 2 and x=2+\sqrt 5. Give your answer in expanded form.

\((x-(1-\sqrt{2}) )(x + 1-\sqrt{2} )(x-(2+\sqrt{5} ))(x+2+\sqrt{5})\)

\((x^2 -(1-\sqrt{2})^2)(x^2 - (2+\sqrt{5})^2)\)

\(x^4 -[(1-\sqrt{2})^2+(2+\sqrt{5})^2]x^2+(1-\sqrt{2})^2(2+\sqrt{5})^2\)

Can you finish it off from there?

As follows:

The ratio of wine to water in bottle A is 5: 3, and the ratio of wine to water in bottle B is 5: 4. Now take the two bottles of solution at 5: 6 and mix them.  What is the ratio of wine to water in the new mixture? 

Find the sum of the x-coordinates of the solutions to the system of equations  y = |x^2-6x+5| and y = (29/4)-x .

Be careful - there are three solutions.  This graph should help you find them.  Simply equating the two functions, but ignoring the absolute value part of the first function will give you two of the solutios.  You need to be more careful to get the solution between x = 1 and x = 5 though.

Answered here: https://web2.0calc.com/questions/please-help_98787#r2 

The polynomial x^4 + 2x^3 - x + 1/4 can be written as the square of a polynomial p(x).  Find p(x).

A fair coin is tossed five times.  What is the probability that heads appears at least once?

First find the probability heads doesn't appear at all:  p_{no_heads} = (1/2)⁵. 

So:   p_{at_least_one_head} = 1 - p_{no_heads}

Use the cosine rule:  \(\cos A=\frac{b^2+c^2-a^2}{2bc}\)

Point C divides the semicircular arc AB of radius 10 in such a way that the product of chords AC and BC is 100.  Find the central angles of the chords.

Not sure if this is what is meant by "central angle"!.  If "central angle" means, say, AOC, where O is the centre of AB, then this would be 180° - 2CAB