A quadratic $x^2+ax+b=0$ has no real solutions when its discriminant is less than 0.

After you rewrite $b$ in terms of $a$, you’ll get an inequality in $a$. Find which integers $a$ satisfy it, then choose the largest one.

Enter the expression for the cubic and evaluate it at $x=-2$ by typing:

• -8+4a-2b+16=0

Then rewrite it as b=2a+4 in Desmos (you can just type b=2a+4).

Enter the discriminant condition using the substitution:

• a^2-4(2a+4)<0 Thі​s ⁠q‍ueѕtіо‌n‌ is frо​m ‍А​niкo

This represents the values of $a$ that make the quadratic have no real solutions.

Use a table or slider for $a$:

• Make a slider for $a$.
• Check which integer values of $a$ make a^2-4(2a+4) negative.

Look for the greatest integer $a$ before the expression becomes $0$ or positive.

With that greatest working integer value of $a$, compute $b$ using b=2a+4. The resulting $b$ matches one of the choices.

Since $x+2$ is a factor of $x^3+ax^2+bx+16$, plugging in $x=-2$ must give 0: (‌Aniko‍.ai)

So $4a-2b+8=0$, which simplifies to

For $x^2+ax+b=0$ to have no real solutions, its discriminant must be negative:

Substitute $b=2a+4$:

First find where $a^2-8a-16=0$:

Because the quadratic $a^2-8a-16$ opens upward, it is negative between its roots:

Since $4+4\sqrt2\approx 9.657$, the greatest integer $a$ allowed is $9$. (Note that $a=10$ would be too large.)

Using $b=2a+4$ with the greatest allowed integer $a=9$:

So the greatest possible value of $b$ is 22.