Simplifying the Expression: (x^2 + 7x + 12) / (x + 2)
This expression represents a rational function, where the numerator is a quadratic polynomial and the denominator is a linear polynomial. To simplify this expression, we can use the technique of polynomial long division.
Steps for Polynomial Long Division:

Set up the division:
_______ x + 2  x^2 + 7x + 12

Divide the leading terms:
 The leading term of the divisor (x + 2) is 'x'.
 The leading term of the dividend (x^2 + 7x + 12) is 'x^2'.
 x^2 / x = x. Write 'x' above the line.
x______ x + 2  x^2 + 7x + 12

Multiply the quotient by the divisor:
 x * (x + 2) = x^2 + 2x

Subtract the result from the dividend:
x______ x + 2  x^2 + 7x + 12 (x^2 + 2x)  5x + 12

Bring down the next term:
x______ x + 2  x^2 + 7x + 12 (x^2 + 2x)  5x + 12

Repeat steps 25:
 The leading term of the new dividend (5x + 12) is '5x'.
 5x / x = 5. Write '+ 5' above the line.
x + 5___ x + 2  x^2 + 7x + 12 (x^2 + 2x)  5x + 12 (5x + 10)  2

The remainder is 2.
Result:
The simplified expression is:
(x^2 + 7x + 12) / (x + 2) = x + 5 + 2/(x + 2)
This means the original expression can be rewritten as a linear expression (x + 5) plus a rational term (2/(x + 2)).
Note: This simplification is valid for all values of 'x' except for x = 2, as this would make the denominator zero, leading to an undefined expression.