15^2 - 63^2 + 27^2 - 22^2 + (54)(15) + (44)(63)=

Key Properties:
#1: \((x + y)^2 = x^2 + 2xy + y^2\)
#2: \((x - y)^2 = x^2 - 2xy + y^2\)
#3: \(x^2-y^2 = (x+y)(x-y)\)

Given: \(15^2 - 63^2 + 27^2 - 22^2 + (54)(15) + (44)(63)\)

Notice that 15 appears in two terms, and 63 appears in two terms.
Also, if we recognize that 54 = (2)(27), we see that 27 appears twice.
And, if we recognize that 44 = (2)(22), we see that 22 appears twice.

Here's what I mean: \(15^2 - 63^2 + 27^2 - 22^2 + (2)(27)(15) + (2)(63)(22)\)

Let's group the similar terms to get: \(27^2 + (2)(27)(15) +15^2 - 63^2 + (2)(63)(22)-22^2\)
Rewrite to get: \([27^2 + (2)(27)(15) +15^2] - [63^2 - (2)(63)(22)+22^2]\)

Notice that \(27^2 + (2)(27)(15) +15^2\) looks a lot like the expansion in key property #1, and \(63^2 - (2)(63)(22)+22^2\) looks a lot like the expansion in key property #2

So, we can factor them as follows: \((27+15)^2 - (63-22)^2\)
Simplify: \(42^2 – 41^2\) [aha! A difference of squares!!]
Use property #3 to factor the expression: \((42+41)(42-41)\)
Simplify: \((83)(1)\)
Evaluate: \(83\)

Answer: B

Cheers,
Brent