Q. \(\,x^2 + 12x + 36\,\)

We want to simplify the expression \(x^2 + 12x + 36\). A common strategy for a quadratic like this is to see whether it can be written as a perfect square.

Step 1: Identify the quadratic form.

The expression is in the form \(ax^2 + bx + c\), where \(a = 1\), \(b = 12\), and \(c = 36\).

Step 2: Check whether it is a perfect square trinomial.

A perfect square trinomial has the form

\[
(x + d)^2 = x^2 + 2dx + d^2.
\]

Step 3: Match coefficients.

We compare \(x^2 + 2dx + d^2\) with \(x^2 + 12x + 36\).

So we need \(2d = 12\).

Step 4: Solve for \(d\).

\[
2d = 12 \quad \Rightarrow \quad d = 6.
\]

Step 5: Check that \(d^2\) matches the constant term.

Compute \(d^2\):

\[
d^2 = 6^2 = 36.
\]

This matches the constant term \(36\), so the trinomial is a perfect square.

Step 6: Rewrite the expression as a perfect square.

\[
x^2 + 12x + 36 = (x + 6)^2.
\]

Final answer: \(\,(x + 6)^2\,\)