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Divide using long division.

\[ \frac{-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6}{x^3 + 6x^2 - 3x - 5} \]

(SHOW WORK)

Answer :

Of course! Let's divide the polynomial [tex]\(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\)[/tex] by [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex] using long division, step-by-step.

### Step 1: Set up the division

Write the division in long division format. The dividend is [tex]\(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6\)[/tex] and the divisor is [tex]\(x^3 + 6x^2 - 3x - 5\)[/tex].

### Step 2: Divide the leading terms

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

[tex]\[
\frac{-3x^5}{x^3} = -3x^2
\][/tex]

This will be the first term of the quotient.

### Step 3: Multiply and subtract

Multiply the entire divisor by [tex]\(-3x^2\)[/tex] and subtract the result from the dividend:

[tex]\[
(-3x^2)(x^3 + 6x^2 - 3x - 5) = -3x^5 - 18x^4 + 9x^3 + 15x^2
\][/tex]

Subtract from the dividend:

[tex]\[
(-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6) - (-3x^5 - 18x^4 + 9x^3 + 15x^2)
\][/tex]

This gives:

[tex]\[
0x^5 - 4x^4 - 22x^3 + 24x^2 + 14x - 6
\][/tex]

### Step 4: Repeat the process

Now take this result as the new dividend and repeat the process.

#### Divide the leading terms:

[tex]\[
\frac{-4x^4}{x^3} = -4x
\][/tex]

Add [tex]\(-4x\)[/tex] to the quotient.

#### Multiply and subtract:

[tex]\[
(-4x)(x^3 + 6x^2 - 3x - 5) = -4x^4 - 24x^3 + 12x^2 + 20x
\][/tex]

Subtract:

[tex]\[
(-4x^4 - 22x^3 + 24x^2 + 14x - 6) - (-4x^4 - 24x^3 + 12x^2 + 20x)
\][/tex]

This gives:

[tex]\[
0x^4 + 2x^3 + 12x^2 - 6x - 6
\][/tex]

### Step 5: Divide again

#### Divide the leading terms:

[tex]\[
\frac{2x^3}{x^3} = 2
\][/tex]

Add [tex]\(2\)[/tex] to the quotient.

#### Multiply and subtract:

[tex]\[
2(x^3 + 6x^2 - 3x - 5) = 2x^3 + 12x^2 - 6x - 10
\][/tex]

Subtract:

[tex]\[
(2x^3 + 12x^2 - 6x - 6) - (2x^3 + 12x^2 - 6x - 10)
\][/tex]

This leaves:

[tex]\[
0x^3 + 0x^2 + 0x + 4 \quad \text{(remainder)}
\][/tex]

### Conclusion

The quotient is [tex]\(-3x^2 - 4x + 2\)[/tex] and the remainder is [tex]\(4\)[/tex]. Therefore, the division can be expressed as:

[tex]\[
\frac{-3x^5 - 22x^4 - 13x^3 + 39x^2 + 14x - 6}{x^3 + 6x^2 - 3x - 5} = -3x^2 - 4x + 2 + \frac{4}{x^3 + 6x^2 - 3x - 5}
\][/tex]

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