Solve the Quadratic Equation: 3x² + 9x - 162 = 0

Let's solve the given equation:

$3x^2+9x-162=0$

$$First, note that the coefficients of all terms in the equation and the free term are divisible by 3 (this can also be determined for the free number - since the sum of its digits is divisible by 3), therefore we will simplify the equation first by dividing both sides by 3:

$3x^2+9x-162=0 \hspace{6pt}\text{/}:3\\ x^2+3x-54=0$

Now we notice that in the resulting equation the coefficient of the squared term is 1, therefore we can (try to) factor the expression on the left side using quick trinomial factoring:

Let's look for a pair of numbers whose product equals the free term in the expression, and whose sum equals the coefficient of the first-degree term, meaning two numbers $m,\hspace{2pt}n$ that satisfy:

$m\cdot n=-54\\ m+n=3\\$From the first requirement mentioned, that is - from the multiplication, we notice that the product of the numbers we're looking for needs to yield a negative result, therefore we can conclude that the two numbers have different signs, according to multiplication rules, and now we'll remember that the number 54 has several possible factor pairs, for example: 27 and 2, 9 and 6, etc. (we won't list them all here) but from the second requirement mentioned, along with the fact that the numbers we're looking for have different signs, leads to the conclusion that the difference between the absolute values of the pair of numbers we're looking for must be 3, therefore the only possibility for the two numbers we're looking for is:

$\begin{cases} m=9\\ n=-6 \end{cases}$

therefore we can factor the expression on the left side of the equation to:

$x^2+3x-54=0 \\ \downarrow\\ (x+9)(x-6)=0$

From here we'll remember that the result of multiplication between expressions will yield 0 only if at least one of the multiplying expressions equals zero,

Therefore we'll get two simple equations and solve them by isolating the unknown on one side:

$x+9=0\\ \boxed{x=-9}$

or:

$x-6=0\\ \boxed{x=6}$

Let's summarize the solution of the equation:

$3x^2+9x-162=0 \\ x^2+3x-54=0 \\ \downarrow\\ (x+9)(x-6)=0 \\ \downarrow\\ x+9=0\rightarrow\boxed{x=-9}\\ x-6=0\rightarrow\boxed{x=6}\\ \downarrow\\ \boxed{x=-9,6}$

Therefore the correct answer is answer D.