Question

1. find the quotient.

options:
$2x^3 - 5x^2 + 10x - 28$
$-2x^2 + 5x - 10$ remainder $28$
$-2x^3 + 5x^2 - 10x + 28$
$2x^2 - 5x + 10$ remainder $-28$

another problem:
$(x^2 - 5x + 2) \div (x - 3)$
options:
$x^2 - 2x - 4$
$x - 2$ remainder $-4$
$-x^2 + 2x + 4$
$x - 2x - 4x$

Explanation:

Step1: Set up synthetic division (Q5)

We divide $2x^3 -5x^2 +10x -28$ by $x+3$ (since divisor is $x-(-3)$).
Coefficients: $2, -5, 10, -28$; root: $-3$

Step2: Bring down leading coefficient

$\downarrow 2$

Step3: Multiply and add (col1)

$2 \times (-3) = -6$; $-5 + (-6) = -11$

Step4: Multiply and add (col2)

$-11 \times (-3) = 33$; $10 + 33 = 43$

Step5: Multiply and add (col3)

$43 \times (-3) = -129$; $-28 + (-129) = -157$
Result: $2x^2 -11x +43$, remainder $-157$ (matches no options? Correction: Wait, original polynomial is $2x^3 -5x^2 +10x -28$, divisor $x+3$:
Wait, recheck: $2(-3)^3 -5(-3)^2 +10(-3)-28 = 2(-27)-5(9)-30-28=-54-45-30-28=-157$. But options have remainder 28: Oh! Wait, divisor is $x-3$? No, table has $-3$. Wait, no—wait the first question's polynomial is $2x^3 -5x^2 +10x -28$ divided by $x+3$, but the option with remainder 28 is $-2x^2+5x-10$ remainder 28: Let's check $(-2x^2+5x-10)(x+3)+28 = -2x^3-6x^2+5x^2+15x-10x-30+28=-2x^3-x^2+5x-2$, not matching. Wait, no—wait the table is

$$\begin{array}{r|rrr} -3 & 2 & -5 & 10 \\ & & & \\ \hline & & & \end{array}$$

(missing constant term? Oh, the polynomial is $2x^3-5x^2+10x$, divided by $x+3$: $2(-3)^3-5(-3)^2+10(-3)=-54-45-30=-129$, no. Wait, the option $2x^3-5x^2+10x-28$: if we divide by $x-2$: $2(8)-5(4)+10(2)-28=16-20+20-28=-12$, no. Wait, correction: The correct synthetic division for question 5:
Wait, the quotient for $2x^3-5x^2+10x-28 \div (x+3)$:

$$\begin{array}{r|rrrr} -3 & 2 & -5 & 10 & -28 \\ & & -6 & 33 & -129 \\ \hline & 2 & -11 & 43 & -157 \end{array}$$

This is not among options, so likely a typo: if divisor is $x-2$:

$$\begin{array}{r|rrrr} 2 & 2 & -5 & 10 & -28 \\ & & 4 & -2 & 16 \\ \hline & 2 & -1 & 8 & -12 \end{array}$$

No. Wait, option 2: $-2x^2+5x-10$ remainder 28: $(-2x^2+5x-10)(x+3)+28 = -2x^3-6x^2+5x^2+15x-10x-30+28=-2x^3-x^2+5x-2$. Not matching. Wait, maybe the polynomial is $-2x^3-5x^2+10x-28$:

$$\begin{array}{r|rrrr} -3 & -2 & -5 & 10 & -28 \\ & & 6 & -3 & -21 \\ \hline & -2 & 1 & 7 & -49 \end{array}$$

No. Wait, question 6: $(x^2-5x+2)\div(x-3)$

Step6: Set up synthetic division (Q6)

Coefficients: $1, -5, 2$; root: $3$

Step7: Bring down leading coefficient

$\downarrow 1$

Step8: Multiply and add (col1)

$1 \times 3 = 3$; $-5 + 3 = -2$

Step9: Multiply and add (col2)

$-2 \times 3 = -6$; $2 + (-6) = -4$
Result: Quotient $x-2$, remainder $-4$

Answer:

1. (Note: There is a discrepancy, but if forced to match, none are correct; assuming a typo, if divisor was $x-2$, no. Wait, recheck: $2x^3-5x^2+10x-28$ divided by $x+3$ gives quotient $2x^2-11x+43$, remainder $-157$. None of the options match. For question 6:
2. $x - 2$ Remainder $-4$