Answer :

We start by simplifying the expression
[tex]$$
n^2 - (n+p)(n-p).
$$[/tex]

First, recall the difference of two squares identity:
[tex]$$
(n+p)(n-p) = n^2 - p^2.
$$[/tex]

Substituting this into the original expression gives:
[tex]$$
n^2 - (n^2 - p^2).
$$[/tex]

Distribute the subtraction:
[tex]$$
n^2 - n^2 + p^2 = p^2.
$$[/tex]

Thus, the expression simplifies to
[tex]$$
p^2.
$$[/tex]

Next, to evaluate
[tex]$$
89\,548^2 - 89\,552 \times 89\,544,
$$[/tex]
we compare it with the simplified form. Notice that the factors in the product correspond to [tex]$n + p = 89\,552$[/tex] and [tex]$n - p = 89\,544$[/tex], while [tex]$n = 89\,548$[/tex]. Therefore, we determine [tex]$p$[/tex] by computing:
[tex]$$
p = 89\,552 - 89\,548 = 4.
$$[/tex]

Now substitute [tex]$p = 4$[/tex] into the simplified expression:
[tex]$$
p^2 = 4^2 = 16.
$$[/tex]

Thus, the final answer is:
[tex]$$
16.
$$[/tex]

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Rewritten by : Barada