Simplify completely: \[ \frac{4+\frac{28}{p}}{1-\frac{49}{p^{2}}} \]

To simplify the expression \(\frac{4+\frac{28}{p}}{1-\frac{49}{p^{2}}}\), we start by rewriting it in a more manageable form. First, the numerator: \[ 4 + \frac{28}{p} = \frac{4p + 28}{p} \] Next, the denominator: \[ 1 - \frac{49}{p^2} = \frac{p^2 - 49}{p^2} = \frac{(p - 7)(p + 7)}{p^2} \] Now we can rewrite the entire expression: \[ \frac{\frac{4p + 28}{p}}{\frac{(p - 7)(p + 7)}{p^2}} = \frac{4p + 28}{p} \cdot \frac{p^2}{(p - 7)(p + 7)} \] This simplifies to: \[ \frac{(4p + 28)p}{(p - 7)(p + 7)} \] Next, we notice that we can factor the numerator: \[ 4p + 28 = 4(p + 7) \] Now substituting this back into our expression gives us: \[ \frac{4(p + 7)p}{(p - 7)(p + 7)} \] Since \(p + 7\) is present in both the numerator and denominator, we can cancel it (as long as \(p \neq -7\)): \[ \frac{4p}{p - 7} \] Thus, the completely simplified form of the original expression is: \[ \frac{4p}{p - 7}, \quad p \neq -7 \] So, the final answer is: \[ \frac{4p}{p - 7}, \quad p \neq -7 \]