To simplify the expression [tex]\(\frac{x^2 - 4x - 21}{4(x+3)}\)[/tex], follow these steps:
### Step 1: Factor the numerator
The numerator is a quadratic expression [tex]\(x^2 - 4x - 21\)[/tex]. To factor this, we need to find two numbers that multiply to [tex]\(-21\)[/tex] and add to [tex]\(-4\)[/tex].
The factors of [tex]\(-21\)[/tex] are:
[tex]\[
1 \cdot (-21), \quad (-1) \cdot 21, \quad 3 \cdot (-7), \quad (-3) \cdot 7
\][/tex]
We need to find a pair that sums to [tex]\(-4\)[/tex]:
[tex]\[
3 + (-7) = -4
\][/tex]
So, we can factor the quadratic expression as:
[tex]\[
x^2 - 4x - 21 = (x + 3)(x - 7)
\][/tex]
### Step 2: Substitute the factored form into the original expression
The original expression becomes:
[tex]\[
\frac{(x + 3)(x - 7)}{4(x + 3)}
\][/tex]
### Step 3: Simplify by canceling out the common term
Since [tex]\(x + 3\)[/tex] appears in both the numerator and the denominator, we can cancel out [tex]\(x + 3\)[/tex] (as long as [tex]\(x \neq -3\)[/tex], because division by zero is undefined):
[tex]\[
\frac{(x + 3)(x - 7)}{4(x + 3)} = \frac{x - 7}{4} \quad \text{for} \quad x \neq -3
\][/tex]
### Conclusion
The simplified form of the expression is:
[tex]\[
\frac{x - 7}{4}
\][/tex]
Thus, the correct choice is:
[tex]\(\boxed{\frac{x-7}{4}}\)[/tex]
