Let's solve the given polynomial division step-by-step.
We need to divide the polynomial \( 10x^4 - 14x^3 - 10x^2 + 6x - 10 \) by the polynomial \( x^3 - 3x^2 + x - 2 \).
### Step 1: Set Up the Division
[tex]\[
\begin{array}{r|rrrrr}
& 10x^4 & -14x^3 & -10x^2 & +6x & -10 \\
\hline
x^3 - 3x^2 + x - 2 & & & & & \\
\end{array}
\][/tex]
We will find the quotient polynomial and the remainder.
### Step 2: Divide the Leading Terms
1. Divide the leading term of the dividend \( 10x^4 \) by the leading term of the divisor \( x^3 \):
[tex]\[
\frac{10x^4}{x^3} = 10x
\][/tex]
The first term of the quotient is \( 10x \).
### Step 3: Multiply and Subtract
2. Multiply \( 10x \) by the entire divisor \( x^3 - 3x^2 + x - 2 \):
[tex]\[
10x \cdot (x^3 - 3x^2 + x - 2) = 10x^4 - 30x^3 + 10x^2 - 20x
\][/tex]
3. Subtract this product from the original polynomial:
[tex]\[
(10x^4 - 14x^3 - 10x^2 + 6x - 10) - (10x^4 - 30x^3 + 10x^2 - 20x) = 16x^3 - 20x^2 + 26x - 10
\][/tex]
### Step 4: Repeat the Process
4. Divide the leading term of the new polynomial \( 16x^3 \) by the leading term of the divisor \( x^3 \):
[tex]\[
\frac{16x^3}{x^3} = 16
\][/tex]
The next term of the quotient is \( 16 \).
### Step 5: Multiply and Subtract Again
5. Multiply \( 16 \) by the entire divisor \( x^3 - 3x^2 + x - 2 \):
[tex]\[
16 \cdot (x^3 - 3x^2 + x - 2) = 16x^3 - 48x^2 + 16x - 32
\][/tex]
6. Subtract this product from the current polynomial:
[tex]\[
(16x^3 - 20x^2 + 26x - 10) - (16x^3 - 48x^2 + 16x - 32) = 28x^2 + 10x + 22
\][/tex]
The polynomial \( 28x^2 + 10x + 22 \) is the remainder since its degree is less than the degree of the divisor.
### Final Result:
Thus, the quotient is \( 10x + 16 \) and the remainder is \( 28x^2 + 10x + 22 \).
So, we can fill in the blanks:
The quotient is \( \boxed{10} \) \( x \) \( + \) \( \boxed{16} \).
The remainder is [tex]\( \boxed{28} \)[/tex] [tex]\( 7x^2 \)[/tex] [tex]\( + \)[/tex] [tex]\( \boxed{10} \)[/tex] [tex]\( x \)[/tex] [tex]\( + \)[/tex] [tex]\( \boxed{22} \)[/tex].
