The problem asks us to factor the expression $64x^2 - 112x + 49$ using the perfect square trinomial formula. We are also told to factor out any common factors.

1. Problem Description

The problem asks us to factor the expression $64x^2 - 112x + 49$ using the perfect square trinomial formula. We are also told to factor out any common factors.

2. Solution Steps

First, we observe that the expression is a quadratic trinomial of the form $ax^2 + bx + c$, where $a=64$, $b=-112$, and $c=49$.

We want to see if this trinomial is a perfect square.

We observe that $64x^2 = (8x)^2$ and $49 = 7^2$. Thus, we can consider if the expression is of the form $(Ax - B)^2 = A^2x^2 - 2ABx + B^2$.

Here, $A = 8$ and $B = 7$, so we can check the middle term $-2ABx = -2(8)(7)x = -112x$. Since this matches the middle term of our expression, we can write $64x^2 - 112x + 49$ as a perfect square.

We have $64x^2 - 112x + 49 = (8x)^2 - 2(8x)(7) + (7)^2$.

Using the formula $(A - B)^2 = A^2 - 2AB + B^2$, we have $(8x - 7)^2 = (8x)^2 - 2(8x)(7) + 7^2 = 64x^2 - 112x + 49$.

Therefore, the factored form is $(8x - 7)^2$.

3. Final Answer

$(8x - 7)^2$