Festival completaly. -a^2-8a-12 square

To factor the quadratic expression \(-a^2 - 8a - 12\) completely, we can follow these steps:<br /><br />1. **Factor out the negative sign**:<br /> \[<br /> -a^2 - 8a - 12 = -(a^2 + 8a + 12)<br /> \]<br /><br />2. **Factor the quadratic expression inside the parentheses**:<br /> We need to find two numbers that multiply to \(12\) (the constant term) and add up to \(8\) (the coefficient of the linear term).<br /><br /> The pairs of factors of \(12\) are:<br /> \[<br /> 1 \cdot 12, \quad 2 \cdot 6, \quad 3 \cdot 4<br /> \]<br /><br /> Among these pairs, \(2\) and \(6\) add up to \(8\):<br /> \[<br /> 2 + 6 = 8<br /> \]<br /><br />3. **Write the factored form**:<br /> Using the numbers \(2\) and \(6\), we can write the quadratic as:<br /> \[<br /> a^2 + 8a + 12 = (a + 2)(a + 6)<br /> \]<br /><br />4. **Include the negative sign**:<br /> Since we factored out a negative sign initially, we need to include it in the final factored form:<br /> \[<br /> -a^2 - 8a - 12 = -(a + 2)(a + 6)<br /> \]<br /><br />So, the completely factored form of \(-a^2 - 8a - 12\) is:<br />\[<br />-(a + 2)(a + 6)<br />\]