Rom

I don't agree that -4 is an excluded value.

\(\dfrac{\dfrac{4}{2x-1}+\dfrac{1}{4}}{x+1}=\\ \dfrac{\dfrac{16+2x-1}{4(2x-1)}}{x+1} = \\ \dfrac{15+2x}{4(2x-1)(x+1)}\)

The balls can sum to odd if they are

(o, o, o), (o, e, e)

There is only 1 way to choose (o,o,o), i.e. (1,3,5)

There are 3 ways to choose (o,e,e), both the two even numbers, 2 and 4, and one of the odd numbers.

Thus there are 4 total ways 3 balls can sum to an odd number.

There are \(\dbinom{5}{3} = 10\) total ways to select 3 balls from the 5.  Thus

\(P[\text{choose 3 balls that sum to an odd number}]=\dfrac{4}{10}=\dfrac 2 5\)

It's not hard to just list these out.

70, 72, 74, 76, 78, 90, 92, 94, 96, 98

so 10 in total.

Thus P[guessing correctly] = 1/10

\(\dfrac{b^2 + 10b + 24}{10b^2+40b} = \\ \dfrac{ b^2 + 10 b + 24}{10b(b+4)} = \\ \dfrac{(b+6)(b+4)}{10b(b+4)}=\\ \dfrac{b+6}{10b}\)

The only excluded value is 

\(b=0\)

Using brute force I come up with

(p,q) = (7,3), (11,3)

\(\text{just complete the square}\\ x^2 - 5x = \\ \\ x^2 - 5x + \dfrac{25}{4} - \dfrac{25}{4} = \\ \\ \left(x-\dfrac 5 2\right)^2 - \dfrac{25}{4}\)

From 5 seconds on google

Advantages: 

Accessibility – a prearranged target group of people willing to help you.

Promptness of Fieldwork – a combination of online research with opt-ins leads to speedy data collection.

Targeted Focus – in most cases, panels are created around interest/purchasing around specific products or services.  So, who better to survey than the people who already have your company in their consideration set?

Efficiency – no need to re-screen participants on key qualification criteria.

Longitudinal Advantages – a large enough panel gives the client the opportunity to track change in behavior over time.

Concerns:

Cost – access to these specialized respondents is not going to come at a general random sample price.  However, in most cases, panels already have an incentive-based give away included in the price, which should trim the overall cost.

Practiced Bias – using panel respondents too much leads to insider knowledge of your process and or product/service research patterns.  If you survey the same panel time and time again, those respondents become less like real consumers and more like informed and trained survey taking dynamos.

Data Quality – in some cases panel respondents are just that, panel respondents.  They don’t only sign-up to help you out, but they sign up for every panel they come across to make a little extra dough.  Again, make sure you routinely scrub your database and remove “suspect” members.

Using the Panel for the Wrong Study – in some market research studies – such as true advertising awareness – you would be better to use a pure random sample rather than a predetermined set of people.

Appropriate Mix – make sure you have the appropriate mix of respondents (demographics, usographics, etc.) that match your specifics before jumping ahead and surveying the group.

In general the number of complex zeros will be equal to the degree of the polynomical.

But it's possible that some of these zeros are repeated.

f(x) here is just a quadratic in x² so we may be able to easily factor it

\(f(x)=x^4+5x^2+6 = \\ (x^2+3)(x^2+2) =\\ (x+i\sqrt{3})(x-i\sqrt{3})(x+i\sqrt{2})(x-i\sqrt{2})\\ \text{and you can see that we have 4 distinct complex roots}\)

\(f(x) = x^4 - x^3 - 2x - 4 \\ \text{I like to quickly check for rational roots before getting fancy}\\ \text{possible roots here are }x = \pm 1, \pm 2, \pm 4 \\ \text{and a quick check shows that }x=-1,~2 \text{ are actual roots}\\ \text{we can then do the polynomial division to find that}\\ f(x) = (x+1)(x-2)(x^2+2)\\ \text{and complete the factoring as}\\ f(x) = (x+1)(x-2)(x+i\sqrt{2})(x - i \sqrt{2})\)

Another way of approaching this is to see if you get lucky when factoring pieces of it.

\(x^4 - x^3-2x-4 = \\ (x^4-4)-(x^3+2x) = \\ (x^2-2)(x^2+2)-x(x^2+2) = \\ (x^2 +2)(x^2 - x-2) =\\ (x^2+2)(x-2)(x+1) = \\ (x+i\sqrt{2})(x-i\sqrt{2})(x-2)(x+1)\\ \text{which is the same as the first answer with the factors listed in different orders}\)

The second method is certainly faster if you get lucky.