Factor out $3$. Polynomial $x^{2}+2x+24$ is not factored since it does not have any rational roots.
$$3\left(x^{2}+2x+24\right)$$
Steps Using the Quadratic Formula
Quadratic polynomial can be factored using the transformation $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$, where $x_{1}$ and $x_{2}$ are the solutions of the quadratic equation $ax^{2}+bx+c=0$.
$$3x^{2}+6x+72=0$$
All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. The quadratic formula gives two solutions, one when $±$ is addition and one when it is subtraction.
Since the square root of a negative number is not defined in the real field, there are no solutions. Quadratic polynomial cannot be factored.
$$3x^{2}+6x+72$$
Steps Using Direct Factoring Method
Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form $x^2+Bx+C=0$.This is achieved by dividing both sides of the equation by $3$
$$x ^ 2 +2x +24 = 0$$
Let $r$ and $s$ be the factors for the quadratic equation such that $x^2+Bx+C=(x−r)(x−s)$ where sum of factors $(r+s)=−B$ and the product of factors $rs = C$
$$r + s = -2 $$ $$ rs = 24$$
Two numbers $r$ and $s$ sum up to $-2$ exactly when the average of the two numbers is $\frac{1}{2}*-2 = -1$. You can also see that the midpoint of $r$ and $s$ corresponds to the axis of symmetry of the parabola represented by the quadratic equation $y=x^2+Bx+C$. The values of $r$ and $s$ are equidistant from the center by an unknown quantity $u$. Express $r$ and $s$ with respect to variable $u$.
$$r = -1 - u$$ $$s = -1 + u$$
To solve for unknown quantity $u$, substitute these in the product equation $rs = 24$
$$(-1 - u) (-1 + u) = 24$$
Simplify by expanding $(a -b) (a + b) = a^2 – b^2$
$$1 - u^2 = 24$$
Simplify the expression by subtracting $1$ on both sides
$$-u^2 = 24-1 = 23$$
Simplify the expression by multiplying $-1$ on both sides and take the square root to obtain the value of unknown variable $u$
