To factor the polynomial expression \(7x^2 + 42x + 63 = 0\), we can first factor out the greatest common factor (GCF) from the expression.
The GCF of \(7x^2\), \(42x\), and \(63\) is \(7\). Thus, we can factor out \(7\):
\[ 7(x^2 + 6x + 9) = 0 \]
Next, we need to factor the quadratic expression \(x^2 + 6x + 9\). This quadratic can be factored as follows:
\[ x^2 + 6x + 9 = (x + 3)(x + 3) = (x + 3)^2 \]
Putting it all together, the original expression factors to:
\[ 7(x + 3)^2 = 0 \]
To solve for \(x\), we set the factored equation equal to zero:
\[ 7(x + 3)^2 = 0 \]
Dividing both sides by \(7\):
\[ (x + 3)^2 = 0 \]
Taking the square root of both sides:
\[ x + 3 = 0 \]
Subtracting \(3\) from both sides gives:
\[ x = -3 \]
Thus, the solution for \(x\) is:
\[ \boxed{-3} \]
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