To solve the equation \( m^2 + 4m + 21 = -6m \), we first rearrange it into standard form.
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Move all terms to one side of the equation: \[ m^2 + 4m + 21 + 6m = 0 \] This simplifies to: \[ m^2 + 10m + 21 = 0 \]
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Next, we need to factor the quadratic equation. We look for two numbers that multiply to \( 21 \) (the constant term) and add to \( 10 \) (the coefficient of \( m \)): The numbers \( 3 \) and \( 7 \) satisfy these conditions since \( 3 \times 7 = 21 \) and \( 3 + 7 = 10 \).
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We can now write the factored form: \[ (m + 3)(m + 7) = 0 \]
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Using the zero product property, set each factor equal to zero: \[ m + 3 = 0 \quad \Rightarrow \quad m = -3 \] \[ m + 7 = 0 \quad \Rightarrow \quad m = -7 \]
Thus, the solutions are: \[ m = -3, -7 \]
So the correct answer is \( m = -3, -7 \).