Certainly! Let's solve the equation step-by-step to find the value of [tex]\( g \)[/tex] that makes it true.
We start with the given equation:
[tex]\[
(x + 7)(x - 4) = x^2 + gx - 28
\][/tex]
To find [tex]\( g \)[/tex], we will first expand the left-hand side of the equation:
[tex]\[
(x + 7)(x - 4)
\][/tex]
Using the distributive property (also known as the FOIL method for binomials):
[tex]\[
(x + 7)(x - 4) = x \cdot x + x \cdot (-4) + 7 \cdot x + 7 \cdot (-4)
\][/tex]
Simplify each term:
[tex]\[
x \cdot x = x^2
\][/tex]
[tex]\[
x \cdot (-4) = -4x
\][/tex]
[tex]\[
7 \cdot x = 7x
\][/tex]
[tex]\[
7 \cdot (-4) = -28
\][/tex]
Now sum all the terms:
[tex]\[
x^2 + (-4x) + 7x + (-28)
\][/tex]
Combine like terms:
[tex]\[
x^2 + (-4x + 7x) - 28
\][/tex]
[tex]\[
x^2 + 3x - 28
\][/tex]
Now we compare this with the original equation:
[tex]\[
x^2 + 3x - 28 = x^2 + gx - 28
\][/tex]
From this comparison, we see that the coefficient of [tex]\( x \)[/tex] on both sides must be equal. Therefore:
[tex]\[
g = 3
\][/tex]
Thus, the value of [tex]\( g \)[/tex] that makes the equation true is:
[tex]\[
\boxed{3}
\][/tex]
