Factor. \[ -20 r^{2}+60 r-45 \] \( -20 r^{2}+60 r-45=\square( \) Factor completely.)

To factor the expression \(-20 r^{2} + 60 r - 45\), we can first factor out the greatest common factor (GCF), which is \(-5\): \[ -5(4r^2 - 12r + 9) \] Next, we can factor the quadratic \(4r^2 - 12r + 9\). This can be factored using the method of finding two numbers that multiply to \(4 \times 9 = 36\) and add to \(-12\). Those numbers are \(-6\) and \(-6\), leading to: \[ 4r^2 - 12r + 9 = (2r - 3)(2r - 3) = (2r - 3)^2 \] So we rewrite the expression: \[ -20 r^{2}+60 r-45 = -5(2r - 3)^2 \] Now we have the complete factorization: \[ -5(2r - 3)^2 \] Feeling like a math detective? It's actually quite fascinating how factoring can sometimes feel like uncovering hidden treasures in equations! It's almost like a magic trick where numbers reappear in a different form, revealing deeper connections between them. For those looking to sharpen their factoring skills, practicing with different types of polynomials can really enhance your abilities. You could start with simple quadratics and soon progress to more complex expressions. There are tons of resources available, including online tutorials and math games that will make the process engaging and rewarding!