Question

fill in the template below to use synthetic division to divide:$(6x^{3}+7x^{2}-12x-10)\div(x-2)$note: only type the sign if the number is negative.result:quotient + remainder = $square$ choose the correct answer from the choices in the table below. type the letter of the correct choice.| | ||---|---|| a | $6x^{2}+5x-2+\frac{6}{x-2}$ || b | $6x^{2}+19x+26+\frac{42}{x-2}$ || c | $6x^{2}-5x-2-\frac{6}{x-2}$ || d | $6x^{2}-19x+26-\frac{42}{x-2}$ |

Explanation:

Step1: Identify synthetic division inputs

For dividing $(6x^3 + 7x^2 - 12x - 10)$ by $(x-2)$, use root $2$, and coefficients $6, 7, -12, -10$.
Set up the table:
Top row (coefficients/root): $2$, $6$, $7$, $-12$, $-10$

Step2: Bring down leading coefficient

Bring down the $6$ to the bottom row.
Bottom row first value: $6$

Step3: Multiply and add (first column)

Multiply $6 \times 2 = 12$, add to $7$: $7 + 12 = 19$
Middle row first value: $12$; Bottom row second value: $19$

Step4: Multiply and add (second column)

Multiply $19 \times 2 = 38$, add to $-12$: $-12 + 38 = 26$
Middle row second value: $38$; Bottom row third value: $26$

Step5: Multiply and add (third column)

Multiply $26 \times 2 = 52$, add to $-10$: $-10 + 52 = 42$
Middle row third value: $52$; Bottom row fourth value: $42$

Step6: Form quotient and remainder

Bottom row values $6, 19, 26$ are coefficients of quotient $6x^2 + 19x + 26$, remainder is $42$.
Result: $6x^2 + 19x + 26 + \frac{42}{x-2}$

Answer:

B. $6x^2 + 19x + 26 + \frac{42}{x-2}$

Filled synthetic division table (row-wise):
Top row: $2$, $6$, $7$, $-12$, $-10$
Middle row: $12$, $38$, $52$
Bottom row: $6$, $19$, $26$, $42$