Factor the expression below completely. All coefficients should be integers.
Ax^2+
Bx+
C
(Ax + E)(x + F)
This expression is in the form \blue{A}x^2 + \green{B}x + \pink{C}.
You can factor it by grouping.
First, find two values, a and
b, so:
\qquad \begin{eqnarray}
\purple{ab} &=& \blue{A}\pink{C} \\
\purple{a} + \purple{b} &=& \green{B}
\end{eqnarray}
In this case:
\qquad \begin{eqnarray}
\purple{ab} &=&
\blue{(A)}\pink{(C)} &=& A * C \\
\purple{a} + \purple{b} &=& & &
\green{B}
\end{eqnarray}
In order to find \purple{a} and \purple{b}, list out the factors of
A * C and add them together.
Remember, since A * C is negative, one of the factors must be
negative.
The factors that add up to \green{B} will be your
\purple{a} and \purple{b}.
When \purple{a} is \purple{E} and
\purple{b} is \purple{A * F}:
\qquad \begin{eqnarray}
\purple{ab} &=& (\purple{E})(\purple{A * F})
&=& E * A * F \\
\purple{a} + \purple{b} &=& \purple{E} + \purple{A * F}
&=& E + A * F
\end{eqnarray}
Next, rewrite the expression as \blue{A}x^2 + \purple{a}x + \purple{b}x + \pink{C}:
\qquad \blue{A}x^2
+\purple{E}x
+\purple{A * F}x
+\pink{C}
Group the terms so that there is a common factor in each group:
\qquad (\blue{A}x^2 +\purple{E}x)
+ (\purple{A * F}x +\pink{C})
Factor out the common factors:
\qquad x(Ax + E) + F(Ax + E)
Notice how (Ax + E) has become a common factor.
Factor this out to find the answer.
(Ax + E)(x + F)