To determine the value of [tex]\( m \)[/tex] such that the polynomials [tex]\( p(x) \)[/tex] and [tex]\( g(x) \)[/tex] are equal, follow these steps:
Given polynomials are:
[tex]\[ p(x) = 8x^2 - 2mx + 5 \][/tex]
[tex]\[ g(x) = 8x^2 - (m + 2)x + 5 \][/tex]
For these polynomials to be equal, their corresponding coefficients must be identical. Let's compare the coefficients of like terms one by one:
1. The coefficients of [tex]\( x^2 \)[/tex]:
[tex]\[ 8 = 8 \][/tex]
This is already satisfied.
2. The coefficients of the constant term:
[tex]\[ 5 = 5 \][/tex]
This is also already satisfied.
3. The coefficients of [tex]\( x \)[/tex]:
We need to equate the coefficients of [tex]\( x \)[/tex] from both polynomials:
[tex]\[ -2m = -(m + 2) \][/tex]
Now, let's solve this equation for [tex]\( m \)[/tex]:
First, remove the negative signs from both sides:
[tex]\[ -2m = -m - 2 \][/tex]
Next, we simplify and rearrange the equation to isolate [tex]\( m \)[/tex] on one side:
[tex]\[ -2m + m = -2 \][/tex]
[tex]\[ -m = -2 \][/tex]
Finally, multiply both sides of the equation by -1 to solve for [tex]\( m \)[/tex]:
[tex]\[ m = 2 \][/tex]
Therefore, the value of [tex]\( m \)[/tex] is:
[tex]\[ \boxed{2} \][/tex]
