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)
]
