Simplifying the Expression (x^3  13x  12) / (x  4)
This expression represents a rational function, which is a fraction where both the numerator and denominator are polynomials. To simplify this expression, we can utilize polynomial long division.
Polynomial Long Division

Set up the division:
________ x  4  x^3 + 0x^2  13x  12

Divide the leading terms:
 The leading term of the divisor (x  4) is x.
 The leading term of the dividend (x^3) is x^3.
 Divide x^3 by x, which gives x^2.
 Write x^2 above the division line.
x^2_______ x  4  x^3 + 0x^2  13x  12

Multiply the quotient by the divisor:
 Multiply x^2 by (x  4), which gives x^3  4x^2.
 Write the result below the dividend.
x^2_______ x  4  x^3 + 0x^2  13x  12 x^3  4x^2

Subtract:
 Subtract the product from the dividend. Remember to change the signs of the terms you are subtracting.
x^2_______ x  4  x^3 + 0x^2  13x  12 x^3  4x^2  4x^2  13x

Bring down the next term:
 Bring down the next term from the dividend (13x).
x^2_______ x  4  x^3 + 0x^2  13x  12 x^3  4x^2  4x^2  13x  12

Repeat steps 25:
 Divide the leading term of the new dividend (4x^2) by the leading term of the divisor (x), which gives 4x.
 Write 4x above the division line.
 Multiply 4x by (x  4), giving 4x^2  16x.
 Subtract this product from the new dividend.
x^2 + 4x_____ x  4  x^3 + 0x^2  13x  12 x^3  4x^2  4x^2  13x  12 4x^2  16x  3x  12

Repeat steps 25 again:
 Divide the leading term of the new dividend (3x) by the leading term of the divisor (x), which gives 3.
 Write 3 above the division line.
 Multiply 3 by (x  4), giving 3x  12.
 Subtract this product from the new dividend.
x^2 + 4x + 3__ x  4  x^3 + 0x^2  13x  12 x^3  4x^2  4x^2  13x  12 4x^2  16x  3x  12 3x  12  0

The result:
 The quotient is x^2 + 4x + 3.
 The remainder is 0.
Therefore, (x^3  13x  12) / (x  4) simplifies to x^2 + 4x + 3.
Conclusion
This simplification demonstrates the effectiveness of polynomial long division for dividing polynomials. The result shows that the original rational function can be expressed as a simpler polynomial, which may be useful in further analysis or calculations.