<P>We clearly know that, for finding a quadratic polynomial with sum of zeroes = S, and product of zeroes = P, we use equation ::

\LARGE\boxed{\bf{p(X) = x^2 - Sx + P}}

p(x)=x

2

−Sx+P

\:

\underline{\sf{\bigstar\:Putting\:all\:known\:values\:::}}

★Puttingallknownvalues::

\begin{gathered}\\ \quad \longrightarrow \quad \sf p(x) = x^2 - \bigg(\dfrac{9}{2}\bigg)x + 2 \end{gathered}

⟶p(x)=x

2

−(

2

9

)x+2

\:

\underline{\sf{\bigstar\:Multiplying\:the\:equation\:with\:2\:::}}

★Multiplyingtheequationwith2::

\begin{gathered}\\ \quad \longrightarrow \quad \sf p(x) = 2\Bigg(x^2 - \bigg(\dfrac{9}{2}\bigg)x + 2\Bigg) \end{gathered}

⟶p(x)=2(x

2

−(

2

9

)x+2)

\begin{gathered}\\ \quad \longrightarrow \quad \sf p(x) = \big(2\:\times\:x^2\big) - \bigg(\dfrac{9}{\cancel{2}}\:\times\:\cancel{2}\bigg)x + \big(2\:\times\:2\big)\end{gathered}

⟶p(x)=(2×x

2

)−(

2

9

×

2

)x+(2×2)

\begin{gathered}\\ \quad \longrightarrow \quad \large \bf 2x^2 - 9x + 4\end{gathered}

⟶2x

2

−9x+4

\:

\therefore\:{\underline{\sf{Hence,\:required\:polynomial\:is\:\bf{2x^2 - 9x + 4}}}}∴

Hence,requiredpolynomialis2x

2

−9x+4

\:

\LARGE\underline{\underline{\textsf{\textbf{Explore\:More\::-}}}}

ExploreMore:-

\:

If α and β are zeroes of the quadratic polynomial ax² + bx + c, then α + β = -b/a and αβ = c/a.

If α, β, γ are the zeroes of cubic polynomial ax³ + bx² + cx + d, then α + β + γ = -b/a, αβ + βγ + γα = c/a and αβγ = -d/a.

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\LARGE\underline{\underline{\textsf{\textbf{Learn\:more\:on\:brainly\::-}}}}

Learnmoreonbrainly:-

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\underline{\sf{\bigstar\:Question\:::}}

★Question::

\:

Find zeroes of the quadratic polynomial 4x² - x - 5 and verify relationship between it's zeroes and coefficients.

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\underline{\sf{\bigstar\:ANSWER\:::}}

★Answer::

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brainly.in/question/429646