The expression 64x^2+56x+k has three terms, as does the right hand side of the equation above.
Equating the two expressions shows that
64x^2 = a^2
56x = 2ab
k = b^2
To find k, we'll need to figure out b.
To figure out b, we need to find 'a' first
If 64x^2 = a^2, then
a^2 = 64x^2
a^2 = (8x)^2
a = 8x
where the last step has us apply the square root to both sides.
We could end up with a = -8x, but this value will ultimately lead to the same value of k. So we'll stick to a = 8x to make things simple.
Use that value of 'a', and the second equation we formed, to get
56x = 2ab
56x = 2(8x)b
56x = 16xb
56 = 16b
16b = 56
b = 56/16
b = (8*7)/(8*2)
b = 7/2
Note: if you went with a = -8x, then b = -7/2. Otherwise, b is positive.
Now we can compute k
k = b^2
k = (7/2)^2
k = (7^2)/(2^2)
k = 49/4
This means 64x^2+56x+k updates to which factors to .
Use the perfect square trinomial formula template, given at the very top of the solution, to help factor.
Because we can rewrite into the form (expression)^2, this proves is a perfect square trinomial.