To determine the value of [tex]\( g \)[/tex] that makes the equation [tex]\( (x + 7)(x - 4) = x^2 + gx - 28 \)[/tex] true, we'll start by expanding the left-hand side of the equation using the distributive property (also known as the FOIL method for binomials).
1. First: Multiply the first terms of each binomial:
[tex]\[
x \cdot x = x^2
\][/tex]
2. Outer: Multiply the outer terms:
[tex]\[
x \cdot (-4) = -4x
\][/tex]
3. Inner: Multiply the inner terms:
[tex]\[
7 \cdot x = 7x
\][/tex]
4. Last: Multiply the last terms of each binomial:
[tex]\[
7 \cdot (-4) = -28
\][/tex]
Next, combine all these products:
[tex]\[
x^2 - 4x + 7x - 28
\][/tex]
Now, combine like terms:
[tex]\[
x^2 + 3x - 28
\][/tex]
We are given that:
[tex]\[
(x + 7)(x - 4) = x^2 + gx - 28
\][/tex]
By comparing the two equations [tex]\( x^2 + 3x - 28 \)[/tex] and [tex]\( x^2 + gx - 28 \)[/tex], we see that the coefficient of [tex]\( x \)[/tex] on the left-hand side (3) must equal the coefficient of [tex]\( x \)[/tex] on the right-hand side ([tex]\( g \)[/tex]).
Therefore, the value of [tex]\( g \)[/tex] that makes the equation true is:
[tex]\[
g = 3
\][/tex]
