Can someone explain how 49c²-112c+64 turns out to be (7c-8)²?

A big part of math is pattern recognition. A lot of homework is just fighting with stuff so you will remember the pattern when you see it again. The most common pattern is (x + a) * (x + b) = x^2 + (a + b)x + ab and the special case (x + a) * (x - a) = x^2 - a^2. When you spot the pattern you can just write the answer from memory.

Now study this page until it seems obvious: en.wikipedia.org/wiki/Polynomial_remainder_theorem

49c^2 - 112c + 64 = 0 First you notice that 49=7^2. Then you notice that 64=8^2. Then you check to see that 2x7x8=112 so you write the answer from memory: (7c - 8)^2 = 0