What is the complete factorisation of \(5x^2-45\)?

Correct answer: \(5(x-3)(x+3)\)

The direct answer is B: \(5x^2-45=5(x-3)(x+3)\). First take out the common factor 5: \(5(x^2-9)\). Now recognize \(x^2-9=x^2-3^2\) as a difference of two squares. Using \(A^2-B^2=(A-B)(A+B)\), it becomes \((x-3)(x+3)\). Therefore the complete factorisation is \(5(x-3)(x+3)\). Option A incorrectly treats 9 as the square root of 9; \((x-9)(x+9)=x^2-81\), not \(x^2-9\). Option C changes the common factor and does not expand to the original expression. Option D takes out 5 but stops too early, because \(x^2-45\) is not the result after correctly extracting 5; the remaining constant is 9. The common error is forgetting that the square root of 9 is 3, not 9. Always factor out the common number first, then inspect the remaining expression.