Q. \[ 486 + 108x + 6x^2 = \]

We want to solve the expression:

\[486 + 108x + 6x^2\;.\]

Usually, when a problem is written with an equals sign at the end, it means we should simplify or rewrite the polynomial in a cleaner form. A common goal is to factor it.

Step 1: Rewrite the polynomial in standard order

The terms already match standard order (constant term, linear term, quadratic term):

\[6x^2 + 108x + 486\;.\]

Step 2: Factor out the greatest common factor

Look at the coefficients \(6\), \(108\), and \(486\). All three are divisible by \(6\):

\[
\frac{108}{6} = 18,\quad \frac{486}{6} = 81.
\]

So factor out \(6\):

\[
6x^2 + 108x + 486 = 6(x^2 + 18x + 81)\;.
\]

Step 3: Recognize the perfect square trinomial

We compare:

\[
x^2 + 18x + 81.
\]

For a perfect square:

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

We need:

\[
2a = 18 \quad \Rightarrow \quad a = 9.
\]

Then:

\[
a^2 = 9^2 = 81,
\]

which matches the constant term. So:

\[
x^2 + 18x + 81 = (x + 9)^2.
\]

Step 4: Write the fully factored form

Substitute back:

\[
6(x^2 + 18x + 81) = 6(x + 9)^2.
\]

Final Answer

\[
486 + 108x + 6x^2 = 6(x + 9)^2.
\]