To factor the quadratic equation \(3m^2 - 9m - 30 = 0\), we can start by factoring out the greatest common factor (GCF) from all terms.
The GCF of \(3m^2\), \(-9m\), and \(-30\) is 3.
First, we factor out 3:
\[ 3(m^2 - 3m - 10) = 0 \]
Next, we need to factor the quadratic expression \(m^2 - 3m - 10\). We are looking for two numbers that multiply to \(-10\) (the constant term) and add to \(-3\) (the coefficient of \(m\)).
The numbers that fulfill this are \(-5\) and \(2\) because:
\[ -5 \times 2 = -10 \quad \text{and} \quad -5 + 2 = -3 \]
Thus, we can factor the quadratic as follows:
\[ m^2 - 3m - 10 = (m - 5)(m + 2) \]
Putting it all together, the equation can now be expressed as:
\[ 3(m - 5)(m + 2) = 0 \]
To solve for \(m\), we set each factor equal to zero:
- \(m - 5 = 0 \Rightarrow m = 5\)
- \(m + 2 = 0 \Rightarrow m = -2\)
The complete factorization of the original equation is:
\[ 3(m - 5)(m + 2) = 0 \]
And the solutions to the equation \(3m^2 - 9m - 30 = 0\) are:
\[ m = 5 \quad \text{and} \quad m = -2 \]
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