Amal's work is correct.
To elaborate, we can break down the polynomial division step-by-step as follows:
1. Step 1: Divide the leading term of the numerator by the leading term of the divisor:
[tex]\[
\frac{-2x^3}{x^2} = -2x
\][/tex]
2. Step 2: Multiply the entire divisor by the quotient term and subtract from the original polynomial:
[tex]\[
(-2x^3 + 11x^2 - 23x + 20) - (-2x^3 + 6x^2 - 8x)
\][/tex]
This simplifies to:
[tex]\[
(-2x^3 + 11x^2 - 23x + 20) + (2x^3 - 6x^2 + 8x) = 5x^2 - 15x + 20
\][/tex]
3. Step 3: Divide the new polynomial by the leading term of the divisor:
[tex]\[
\frac{5x^2}{x^2} = 5
\][/tex]
4. Step 4: Multiply the entire divisor by the second quotient term and subtract:
[tex]\[
(5x^2 - 15x + 20) - (5x^2 - 15x + 20) = 0
\][/tex]
Since the remainder is zero, the division is complete.
5. Combining the quotients, we get:
[tex]\[
-2x + 5
\][/tex]
Since both intermediate and final steps show the operations were performed correctly, we can conclude:
[tex]\[
\frac{-2 x^3+11 x^2-23 x+20}{x^2-3 x+4} = -2x + 5
\][/tex]
Therefore, Amal's work is indeed correct.
