Alan

Well, I should really add a warning here.  The method is not always very useful.  The danger is that, in general, by the time you do the various substitutions to isolate a, b or c, all you get is the original equation with x replaced by a, say!!

In this case it was fairly easy to guess that 1, 2 and 3 were involved, because their product had to be 6 (the other equations help determine the signs), and I started by assuming they would be integers (I reasoned the question probably wouldn't have been posed if they weren't).  Luckily, this reasoning panned out; however, in general it won't and one could guess forever without getting anywhere!

I've given you a thumbs up, zegroes!

Using p1*v1 = p2*v2 we have  p1*20 = 12.2*2.5 so p1 = 12.2*2.5/20

$${\mathtt{p1}} = {\frac{{\mathtt{12.2}}{\mathtt{\,\times\,}}{\mathtt{2.5}}}{{\mathtt{20}}}} \Rightarrow {\mathtt{p1}} = {\mathtt{1.525}}$$ atm

(x + 3)*(x - 1)*(x + 2) = x³ +4x² + x - 6

To find this, start by assuming (x-a)*(x-b)*(x-c) =  x³ +4x² + x - 6

The left-hand side can be expanded, to get: x³ - (a+b+c)*x² +(a*b+a*c+b*c)*x -a*b*c =  x³ +4x² + x - 6

By equating the coefficients of powers of x on both sides of this equation you get three simultaneous equations in a, b and c, which when solved, produce the result shown above.

By equating the coefficients I mean:

coeff of x²: -(a+b+c) = 4

coeff of x¹: a*b + a*c +b*c = 1

coeff of x⁰: -a*b*c = -6

No, it isn't possible in general.  The diagram below shows two right-angled triangles, both of which have a hypotenuse of length √2.  it is clear that they have different other sides and angles.

N = n + 6                             ...(1)   Where N is the bigger number and n the smaller.

3*n + 4 = 2*N                     ...(2)

Substitute N from eqn (1) into eqn (2)

3*n + 4 = 2*(n + 6)

3*n + 4 = 2*n + 12

3*n -2*n = 12 - 4

n = 8

Substitute this back into eqn(1)

N = 8 + 6

N = 14

To convert from degrees to radians multiply by pi/180.

$${\frac{{\mathtt{225}}{\mathtt{\,\times\,}}{\mathtt{\pi}}}{{\mathtt{180}}}} = {\mathtt{3.926\: \!990\: \!816\: \!987\: \!241\: \!5}}$$

So 225° ≈ 3.927 radians

To turn a fraction into a percentage, just multiply by 100.  

(87/100)*100 = 87, so the answer is 87%

The ratio of width to length is the same for both, so

W/15 = 7.25/8.5  or  W = 15*7.25/8.5 ft

$${\mathtt{W}} = {\frac{{\mathtt{15}}{\mathtt{\,\times\,}}{\mathtt{7.25}}}{{\mathtt{8.5}}}} \Rightarrow {\mathtt{W}} = {\mathtt{12.794\: \!117\: \!647\: \!058\: \!823\: \!5}}$$ ft

So the width of the room is approximately 12.8 feet.

Look at the graph below.  The red triangle is in the fourth quadrant, where sin is negative, cos is positive and tan is negative.  With this in mind we can see that

$$\\
\sin(\frac{7\pi}{4})=-\sin(\frac{\pi}{4})=-\frac{1}{\sqrt2}\\
\cos(\frac{7\pi}{4})=\cos(\frac{\pi}{4})=\frac{1}{\sqrt2}\\
\tan(\frac{7\pi}{4})=-\tan(\frac{\pi}{4})=-1$$