Let's start by factoring the polynomial expression \( 7x^2 + 42x + 63 = 0 \).
First, we can factor out the greatest common factor (GCF), which is 7:
\[ 7(x^2 + 6x + 9) = 0 \]
Now, we can focus on the quadratic expression \( x^2 + 6x + 9 \). We will factor this expression. We can rewrite it as:
\[ x^2 + 6x + 9 = (x + 3)(x + 3) = (x + 3)^2 \]
So the equation now looks like this:
\[ 7(x + 3)^2 = 0 \]
Next, we can set this equation to zero:
\[ (x + 3)^2 = 0 \]
To solve for \( x \), we take the square root of both sides:
\[ x + 3 = 0 \]
Subtracting 3 from both sides gives us:
\[ x = -3 \]
Thus, the solution to the equation \( 7x^2 + 42x + 63 = 0 \) is:
\[ \boxed{-3} \]
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