Factor the expression by grouping. First, the expression needs to be rewritten as $7m^{2}+am+bm+8$. To find $a$ and $b$, set up a system to be solved.
$$a+b=-18$$ $$ab=7\times 8=56$$
Since $ab$ is positive, $a$ and $b$ have the same sign. Since $a+b$ is negative, $a$ and $b$ are both negative. List all such integer pairs that give product $56$.
Rewrite $7m^{2}-18m+8$ as $\left(7m^{2}-14m\right)+\left(-4m+8\right)$.
$$\left(7m^{2}-14m\right)+\left(-4m+8\right)$$
Factor out $7m$ in the first and $-4$ in the second group.
$$7m\left(m-2\right)-4\left(m-2\right)$$
Factor out common term $m-2$ by using distributive property.
$$\left(m-2\right)\left(7m-4\right)$$
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 $ax^{2}+bx+c=0$.
$$7m^{2}-18m+8=0$$
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.
Now solve the equation $m=\frac{18±10}{14}$ when $±$ is plus. Add $18$ to $10$.
$$m=\frac{28}{14}$$
Divide $28$ by $14$.
$$m=2$$
Now solve the equation $m=\frac{18±10}{14}$ when $±$ is minus. Subtract $10$ from $18$.
$$m=\frac{8}{14}$$
Reduce the fraction $\frac{8}{14}$ to lowest terms by extracting and canceling out $2$.
$$m=\frac{4}{7}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $2$ for $x_{1}$ and $\frac{4}{7}$ for $x_{2}$.
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 $7$
$$x ^ 2 -\frac{18}{7}x +\frac{8}{7} = 0$$
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 $rs = C$
$$r + s = \frac{18}{7} $$ $$ rs = \frac{8}{7}$$
Two numbers $r$ and $s$ sum up to $\frac{18}{7}$ exactly when the average of the two numbers is $\frac{1}{2}*\frac{18}{7} = \frac{9}{7}$. 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 $u$.
$$r = \frac{9}{7} - u$$ $$s = \frac{9}{7} + u$$
To solve for unknown quantity $u$, substitute these in the product equation $rs = \frac{8}{7}$
