To factor the expression \(150y^2 - 60y + 6\), we can start by looking for a common factor in all the terms.
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Identify the GCF (greatest common factor) of the coefficients \(150\), \(-60\), and \(6\). The GCF is \(6\).
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Factor out \(6\) from the entire expression:
\[ 150y^2 - 60y + 6 = 6(25y^2 - 10y + 1) \]
Now, we need to factor the quadratic \(25y^2 - 10y + 1\). We can look for two numbers that multiply to \(25 \times 1 = 25\) (the product of the coefficient of \(y^2\) and the constant term) and add to \(-10\) (the coefficient of \(y\)).
The two numbers that satisfy this are \(-5\) and \(-5\) (since \(-5 \times -5 = 25\) and \(-5 + -5 = -10\)). This means we can rewrite the quadratic as:
\[ 25y^2 - 10y + 1 = (5y - 1)(5y - 1) = (5y - 1)^2 \]
Therefore, substituting this back into our factored expression yields:
\[ 150y^2 - 60y + 6 = 6(5y - 1)^2 \]
Thus, the completely factored form of the expression is:
\[ \boxed{6(5y - 1)^2} \]