Q. \(x^2+14x+49\)

We want to simplify factor the quadratic expression \(x^2+14x+49\).

Step 1: Identify the pattern

The expression \(x^2+14x+49\) looks like a perfect square trinomial of the form

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

When a quadratic matches this pattern, it factors as

\[
x^2+2ax+a^2=(x+a)^2
\]

Step 2: Match coefficients

Compare \(x^2+14x+49\) with \(x^2+2ax+a^2\).

\[
2a = 14
\]

Divide both sides by \(2\):

\[
a = 7
\]

Step 3: Verify the constant term

Compute \(a^2\):

\[
a^2 = 7^2 = 49
\]

This matches the constant term \(49\), so the expression is indeed a perfect square.

Step 4: Write the factored form

Substitute \(a=7\) into \((x+a)^2\):

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

Final Answer

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