Q. What is the quotient of \(x^{3} + 3 x^{2} – 4 x – 12,\) divided by \(x^{2} + 5 x + 6\)?

1. Write the dividend and divisor.Dividend: \[x^{3}+3x^{2}-4x-12\]Divisor: \[x^{2}+5x+6\]
2. Observe degrees and factor the divisor (optional).The divisor is degree 2, so the quotient will be degree 1. Factor the divisor to check structure:\[x^{2}+5x+6=(x+2)(x+3)\]
3. First step of polynomial long division: divide leading terms.Divide the leading term of the dividend by the leading term of the divisor:\[\frac{x^{3}}{x^{2}}=x\]

   Multiply the divisor by this term:

   \[x\cdot(x^{2}+5x+6)=x^{3}+5x^{2}+6x\]

   Subtract this product from the dividend (perform subtraction termwise):

   \[
   (x^{3}+3x^{2}-4x-12)-(x^{3}+5x^{2}+6x)
   = (3x^{2}-5x^{2})+(-4x-6x)+(-12)
   = -2x^{2}-10x-12
   \]

4. Second step: divide the new leading term.Divide the new leading term by the divisor’s leading term:\[\frac{-2x^{2}}{x^{2}}=-2\]

   Multiply the divisor by -2:

   \[-2\cdot(x^{2}+5x+6)=-2x^{2}-10x-12\]

   Subtract this from the intermediate remainder:

   \[
   (-2x^{2}-10x-12)-(-2x^{2}-10x-12)=0
   \]

   The remainder is zero, so the division terminates.

5. Conclude the quotient.The quotient is the sum of the terms found in the division steps:\[x+(-2)=x-2\]
6. Check by multiplication (verification).Multiply the divisor by the quotient to verify we recover the dividend:\[
   (x^{2}+5x+6)(x-2)
   = x^{3}+5x^{2}+6x-2x^{2}-10x-12
   = x^{3}+3x^{2}-4x-12
   \]

   The product equals the original dividend, so the quotient is confirmed.

7. Note on the appended expressionIf the expression appended to the problem is to be simplified: \[x+2x-2x-8x+8\]Simplify termwise: \[x+(2x-2x)-8x+8=x-8x+8=-7x+8\]

Final answer (quotient): \[x-2\]

Simplified appended expression (if required): \[-7x+8\]