Question

which shows the four-term polynomial and factored form of $x^2 + 6x - 27$?
$\circ\\ x^2 + 3x - 9x - 27 = (x + 3)(x - 9)$
$\circ\\ x^2 + 6x - 3x - 27 = (x + 6)(x - 3)$
$\circ\\ x^2 + 9x - 3x - 27 = (x + 9)(x - 3)$
$\circ\\ x^2 + 3x - 6x - 27 = (x + 3)(x - 6)$

Explanation:

Step1: Analyze the middle term splitting

For the quadratic \(x^2 + 6x - 27\), we need two numbers that multiply to \(-27\) and add to \(6\). The numbers are \(9\) and \(-3\) (since \(9\times(-3)=-27\) and \(9 + (-3)=6\)).
So, split the middle term: \(x^2 + 9x - 3x - 27\).

Step2: Factor by grouping

Group the terms: \((x^2 + 9x) + (-3x - 27)\).
Factor out common factors: \(x(x + 9) - 3(x + 9)\).
Then, factor out \((x + 9)\): \((x + 9)(x - 3)\).

Step3: Check other options

• First option: \((x + 3)(x - 9)=x^2 - 6x - 27

eq x^2 + 6x - 27\).

• Second option: \((x + 6)(x - 3)=x^2 + 3x - 18

eq x^2 + 6x - 27\).

• Fourth option: \((x + 3)(x - 6)=x^2 - 3x - 18

eq x^2 + 6x - 27\).

Answer:

\(x^2 + 9x - 3x - 27 = (x + 9)(x - 3)\) (the third option)