Determine quadratic factor: u^{2}+7u+6

Step 1
Given: ${\displaystyle u^2+7u+6}$
We have to find integers b and d such that
${\displaystyle u^2+7u+6=(u+b)(u+d)}$
${\displaystyle =u^2+du+bu+bd}$
${\displaystyle u^2+7u+6=u^2+(b+d)u+bd}$
SInce: $bd=6$ b and d are factors of 6
Similarly: ${\displaystyle b+d=7}$
Table:
$$\begin{array}{|cc|}\hline \text{Factor b, d of 6}& \text{Sum}b+d=7\\ 3\times 2& 3+2=5\\ 1\times 6& 1+6=7\\ \hline\end{array}$$
To check:
${\displaystyle (u+1)(u+6)=u(u+6)+1(u+6)}$
${\displaystyle u^2+6u+u+6}$
${\displaystyle =u^2+7y+6}$

Step 1
Let's factor ${\displaystyle u^2+7u+6}$
The middle number is 7 and the last number is 6.
Factoring means we want something like
$(u+\underset{―}{})(u+\underset{―}{})$
Which numbers go in the blanks?
We need two numbers that
Add together to get 7
Multiply together to get 6
Can you think of the two numbers?
Try 1 and 6
${\displaystyle 1+6=7}$
${\displaystyle 1\times 6=6}$
Fill in the blanks in
$(u+\underset{―}{})(u+\underset{―}{})$
with 1 and 6 to get
${\displaystyle (u+1)(u+6)}$

Step 1
Quadratic polynomial can be factored using the transformation $a{x}^2+bx+c=a(x-x_1)(x-x_2)$, where $x_1$ and $x_2$ are the solutions of the quadratic equation $a{x}^2+bx+c=0$
$u^2+7u+6=0$
All equations of the form $a{x}^2+bx+c=0$ can be solved using the quadratic formula $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$. The quadratic formula gives two solutions, one when $\pm$ is addition and one when it is subtraction.
$u=\frac{-7\pm \sqrt{7^2-4\times 6}}{2}$
Square 7
$u=\frac{-7\pm \sqrt{49-4\times 6}}{2}$
Multiply -4 times 6.
$u=\frac{-7\pm \sqrt{49-24}}{2}$
Add 49 to -24
$u=\frac{-7\pm \sqrt{25}}{2}$
Take the square root of 25.
$u=\frac{-7\pm 5}{2}$
Now solve the equation $u=\frac{-7\pm 5}{2}$ when $\pm$ is plus. Add -7 to 5
$u=\frac{-2}{2}$
Divide -2 by 2
u=-1
Now solve the equation $u=\frac{-7\pm 5}{2}$ when $\pm$ is minus. Subtract 5 from -7
u=$\frac{-12}{2}$
Divide -12 by 2
u=-6
Factor the original expression using ax${}^2$+bx+c=a(x-x${}_1$)(x-x${}_2$). Substitute -1 for x${}_1$ and -6 for x${}_2$
u${}^2$+7u+6=(u-(-1))(u-(-6))
Simplify all the expressions of the form p-(-q) to p+q
u${}^2$+7u+6=(u+1)(u+6)
(u+1)(u+6)