x2+x-12 ----------- x2+8x+16

To simplify the expression \(\frac{x^2 + x - 12}{x^2 + 8x + 16}\), we need to factor both the numerator and the denominator, if possible.

Step 1: Factor the numerator \(x^2 + x - 12\)

To factor \(x^2 + x - 12\), we need to find two numbers that multiply to \(-12\) and add to \(1\). These numbers are \(4\) and \(-3\).

So, we can factor the numerator:

\[ x^2 + x - 12 = (x + 4)(x - 3) \]

Step 2: Factor the denominator \(x^2 + 8x + 16\)

To factor \(x^2 + 8x + 16\), we look for two numbers that multiply to \(16\) and add to \(8\). These numbers are \(4\) and \(4\).

So, we can factor the denominator:

\[ x^2 + 8x + 16 = (x + 4)(x + 4) = (x + 4)^2 \]

Step 3: Rewrite the expression with the factors

Now we can rewrite the original expression with the factored forms:

\[ \frac{x^2 + x - 12}{x^2 + 8x + 16} = \frac{(x + 4)(x - 3)}{(x + 4)(x + 4)} \]

Step 4: Simplify the expression

We can cancel out the \((x + 4)\) term from the numerator and the denominator (assuming \(x \neq -4\) since it would lead to division by zero):

\[ = \frac{x - 3}{x + 4} \quad (x \neq -4) \]

Thus, the simplified form of the expression is

\[ \frac{x - 3}{x + 4} \quad (x \neq -4) \]