Q. \(8x^2+112x+392=\)

We want to solve the equation

\[
8x^2 + 112x + 392 = 0
\]

Step 1: Divide every term by the greatest common factor.

All coefficients \(8\), \(112\), and \(392\) are divisible by \(8\). So we divide the entire equation by \(8\).

\[
\frac{8x^2}{8} + \frac{112x}{8} + \frac{392}{8} = \frac{0}{8}
\]

Now simplify each term:

\[
x^2 + 14x + 49 = 0
\]

Step 2: Recognize a perfect square trinomial.

The expression \(x^2 + 14x + 49\) matches the pattern

\[
x^2 + 2ax + a^2
\]

where \(2a = 14\) and \(a^2 = 49\).

Solve \(2a = 14\):

\[
a = 7
\]

Then \(a^2 = 7^2 = 49\), which matches the constant term, so the trinomial is a perfect square:

\[
x^2 + 14x + 49 = (x + 7)^2
\]

Step 3: Substitute and solve.

Replace the left side with \((x+7)^2\):

\[
(x+7)^2 = 0
\]

Step 4: Take the square root of both sides.

If \((x+7)^2 = 0\), then the only way to get \(0\) is:

\[
x+7 = 0
\]

Step 5: Solve for \(x\).

\[
x = -7
\]

Final Answer:

\[
\boxed{x = -7}
\]