the constant differences of an 8 degree polynomial is 241920.What is the leading coefficient?

Find the point on the line $y=5x+2$ that is closest to the origin.

5)The graph of f is given. State the numbers at which f is not differentiable.
x=?(smaller value)
x=?(larger value)

6)The graph of f is given. State the numbers at which f is not differentiable.
x=?(smaller value)
x=?(larger value)

Discriminant Problems
We all know that the discriminant is the part ${\displaystyle b^2-4ac}$ of the equation
${\displaystyle x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}}$
that we use to find the roots of a quadratic equation eg:
${\displaystyle a{x}^2+bx+c=0}$ or the part in a trinomial expression like ${\displaystyle a{x}^2+bx+c}$.
What I want to know is what kind of ideas we can conclude for a given inequality of a discriminant. We know that in quadratic formulas the number of real roots depends on the case whether the discriminant is <0, 0, or >0. But if given only the discriminant and the inequality of it, for examples like this:
${\displaystyle b^2-4ac=0}$
${\displaystyle b^2-4ac>0}$
${\displaystyle b^2-4ac<0}$
What can we conclude by the given information for each case? How can it help to reveal certain problems involving trinomial expressions?

Let $X(t),A,B,C$ be matrices and $A,B,C$ are constant matrices. Does the following linear differential equation have a closed form solution?
$\frac{dX(t)}{dt}=A+BXC.$

What is the value of ${\displaystyle x-y}$, if ${\displaystyle xy=144}$, ${\displaystyle x+y=30}$, and ${\displaystyle x>y}$?

For an irrational number
$x=\sum _{n\in \mathbb{Z}}{a}_n{10}^n$
Does $a_n$ takes all the numbers between $0$ and $9$ ?

Given a system of inequalities $Ax\le b$ , how can I derive the upper and lower bound on $x$?
I have the following system of inequalities $Ax\le b$, where $x_1$ and $x_2$ are unknown, and the $a$'s and $и$'s are constants.
$\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]\le \left[\begin{array}{c}{b}_1\\ b_2\end{array}\right]$