Ques 4. (Plz show all possible steps with explanation )

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{We} \mathrm{have} \\ {\mathrm{ax}}^2+\mathrm{bx}+8=0 \\ \mathrm{Roots} \mathrm{are} \mathrm{not} \mathrm{real} \mathrm{and} \mathrm{distinct}. \mathrm{Hence} \\ \mathrm{Discriminant}\le 0 \\ \Rightarrow {\mathrm{b}}^2-4\times \mathrm{a}\times 8\le 0 \\ \Rightarrow {\mathrm{b}}^2-32\mathrm{a}\le 0 \\ \Rightarrow \mathrm{a}\ge \frac{{\mathrm{b}}^2}{32} \\ \mathrm{So} \mathrm{we} \mathrm{have} \\ 4\mathrm{a}+\mathrm{b}\ge 4\left(\frac{{\mathrm{b}}^2}{32}\right)+\mathrm{b} \\ \ge \frac{{\mathrm{b}}^2}{8}+\mathrm{b} \\ \ge \frac{\left({\mathrm{b}}^2+8\mathrm{b}\right)}{8} \\ 4\mathrm{a}+\mathrm{b}\ge \frac{{\left(\mathrm{b}+4\right)}^2-16}{8} \\ \mathrm{Least} \mathrm{value} \mathrm{is} \mathrm{obtained} \mathrm{when} {\left(\mathrm{b}+4\right)}^2 \mathrm{is} 0 \\ \Rightarrow {\left[4\mathrm{a}+\mathrm{b}\right]}_{\min }=\frac{-16}{8}=-2 \left[\mathrm{Answer}\right] \end{gathered}$$

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