Answer :

To find the quotient and remainder when the polynomial [tex]\(2x^2 - 5x + 3\)[/tex] is divided by [tex]\(x\)[/tex], we can use polynomial long division. Let's go through the steps together:

1. Set up the division. Write the dividend [tex]\(2x^2 - 5x + 3\)[/tex] under the long division symbol, and the divisor [tex]\(x\)[/tex] outside:

[tex]\[ \begin{array}{r|lr} & 2x^2 - 5x + 3 \\ x & \end{array} \][/tex]

2. Divide the first term of the dividend by the first term of the divisor:

[tex]\[ \frac{2x^2}{x} = 2x \][/tex]

This gives us the first term of the quotient, [tex]\(2x\)[/tex].

3. Multiply the entire divisor [tex]\(x\)[/tex] by the first term of the quotient [tex]\((2x)\)[/tex] and write the result below the corresponding terms of the dividend:

[tex]\[ 2x \cdot x = 2x^2 \][/tex]

[tex]\[ \begin{array}{r|lr} & 2x^2 - 5x + 3 \\ x & 2x^2 \end{array} \][/tex]

4. Subtract this result from the original dividend:

[tex]\[ (2x^2 - 5x + 3) - (2x^2) = -5x + 3 \][/tex]

So now, we write the intermediate result:

[tex]\[ \begin{array}{r|lr} & 2x^2 - 5x + 3 \\ x & 2x^2 \\ & \underline{- 2x^2} \\ & - 5x + 3 \\ \end{array} \][/tex]

5. Divide the first term of the new intermediate result by the first term of the divisor:

[tex]\[ \frac{-5x}{x} = -5 \][/tex]

This gives us the next term of the quotient, [tex]\(-5\)[/tex].

6. Multiply the entire divisor [tex]\(x\)[/tex] by the next term of the quotient [tex]\((-5)\)[/tex] and write the result below the corresponding terms of the intermediate result:

[tex]\[ -5 \cdot x = -5x \][/tex]

[tex]\[ \begin{array}{r|lr} & 2x^2 - 5x + 3 \\ x & 2x^2 - 5 \\ & \underline{- 2x^2} \\ & - 5x + 3 \\ & \underline{- 5x} \\ \end{array} \][/tex]

7. Subtract this result from the intermediate result:

[tex]\[ (-5x + 3) - (-5x) = 3 \][/tex]

This gives the new intermediate result:

[tex]\[ \begin{array}{r|lr} & 2x^2 - 5x + 3 \\ x & 2x^2 - 5 \\ & \underline{- 2x^2} \\ & - 5x + 3 \\ & \underline{- 5x} \\ & 3 \\ \end{array} \][/tex]

8. Since we cannot divide 3 by [tex]\(x\)[/tex] (as 3 is a constant and [tex]\(x\)[/tex] is a variable), 3 becomes the remainder.

Therefore, the quotient is [tex]\(2x - 5\)[/tex] and the remainder is [tex]\(3\)[/tex]. So, our final answer is:

[tex]\[ \boxed{(2x - 5, \text{ remainder } 3)} \][/tex]