Question

Factor completely.
\[8 x^2 +112 x +392=\]

Answer 1

To factor the expression \( 8x^2 + 112x + 392 \), we can start by factoring out the greatest common factor (GCF) from all the terms.

1. Identify the GCF of the coefficients:
- The coefficients are 8, 112, and 392.
- The GCF is \( 8 \).

2. Factor out \( 8 \):
\[
8x^2 + 112x + 392 = 8(x^2 + 14x + 49)
\]

3. Now, we will focus on factoring the quadratic \( x^2 + 14x + 49 \).
- To factor this, we can look for two numbers that multiply to \( 49 \) (the constant term) and add to \( 14 \) (the coefficient of \( x \)).
- The numbers \( 7 \) and \( 7 \) satisfy this because \( 7 \times 7 = 49 \) and \( 7 + 7 = 14 \).

4. Thus, we can rewrite the quadratic:
\[
x^2 + 14x + 49 = (x + 7)(x + 7) = (x + 7)^2
\]

5. Now substitute this back into our expression:
\[
8(x^2 + 14x + 49) = 8((x + 7)^2) = 8(x + 7)^2
\]

Hence, the fully factored form of the original expression is:
\[
\boxed{8(x + 7)^2}
\]