Let's begin by simplifying the given polynomial [tex]\( 8x^2 - 8x + 2 - 5 + x \)[/tex].
1. Combine like terms in the polynomial:
- The [tex]\( x^2 \)[/tex] term is [tex]\( 8x^2 \)[/tex].
- The [tex]\( x \)[/tex] terms are [tex]\( -8x \)[/tex] and [tex]\( x \)[/tex], which combine to be [tex]\( -8x + x = -7x \)[/tex].
- The constant terms are [tex]\( 2 \)[/tex] and [tex]\( -5 \)[/tex], which combine to be [tex]\( 2 - 5 = -3 \)[/tex].
So, the simplified polynomial is:
[tex]\[ 8x^2 - 7x - 3 \][/tex]
2. We need to compare this simplified polynomial to the form [tex]\( 8x^2 - gx - h \)[/tex]. From the comparison, we can easily identify the coefficients of [tex]\( x \)[/tex] and the constant term.
Thus:
- The coefficient of [tex]\( x \)[/tex], which is [tex]\( -g \)[/tex], must match the coefficient of [tex]\( x \)[/tex] from the simplified polynomial, [tex]\( -7 \)[/tex]. Therefore, [tex]\( g = 7 \)[/tex].
- The constant term, which is [tex]\( -h \)[/tex], must match the constant term from the simplified polynomial, [tex]\( -3 \)[/tex]. Therefore, [tex]\( h = 3 \)[/tex].
Hence, the values are:
[tex]\[ g = 7 \][/tex]
[tex]\[ h = 3 \][/tex]
So the correct answer is:
[tex]\[ g = 7 \text{ and } h = 3 \][/tex]
