Factor completely. -5x^2+35x+90 square square

To factor the quadratic expression \(-5x^2 + 35x + 90\) completely, we can follow these steps:<br /><br />1. **Factor out the greatest common factor (GCF):**<br /> The GCF of the coefficients \(-5\), \(35\), and \(90\) is \(5\). However, since the leading coefficient is negative, we should factor out \(-5\):<br /> \[<br /> -5(x^2 - 7x - 18)<br /> \]<br /><br />2. **Factor the quadratic expression inside the parentheses:**<br /> We need to factor \(x^2 - 7x - 18\). We look for two numbers that multiply to \(-18\) (the constant term) and add up to \(-7\) (the coefficient of the linear term).<br /><br /> The numbers that satisfy these conditions are \(-9\) and \(2\), because:<br /> \[<br /> \times 2 = -18 \quad \text{and} \quad -9 + 2 = -7<br /> \]<br /><br />3. **Write the factored form:**<br /> Using the numbers \(-9\) and \(2\), we can write the quadratic as:<br /> \[<br /> x^2 - 7x - 18 = (x - 9)(x + 2)<br /> \]<br /><br />4. **Include the GCF factored out earlier:**<br /> Finally, we include the \(-5\) we factored out initially:<br /> \[<br /> -5(x - 9)(x + 2)<br /> \]<br /><br />So, the completely factored form of \(-5x^2 + 35x + 90\) is:<br />\[<br />\boxed{-5(x - 9)(x + 2)}<br />\]