To determine which of the given expressions has the factor [tex]\((h + 2)\)[/tex], we will factor each expression and see which one includes [tex]\((h + 2)\)[/tex] as a factor.
1. Expression A: [tex]\(2h^2 + h\)[/tex]
2. Expression B: [tex]\(h^2 + 6h + 5\)[/tex]
3. Expression C: [tex]\(2h^2 - 4h\)[/tex]
4. Expression D: [tex]\(h^2 + h - 2\)[/tex]
Let's factorize each expression step-by-step:
Expression A: [tex]\(2h^2 + h\)[/tex]
Factor out the greatest common factor (GCF):
[tex]\[ 2h^2 + h = h(2h + 1) \][/tex]
The factors are [tex]\(h\)[/tex] and [tex]\((2h + 1)\)[/tex]. No factor of [tex]\((h + 2)\)[/tex] here.
Expression B: [tex]\(h^2 + 6h + 5\)[/tex]
To factor the quadratic expression, we need to find two numbers that multiply to [tex]\(5\)[/tex] and add up to [tex]\(6\)[/tex]. These numbers are [tex]\(1\)[/tex] and [tex]\(5\)[/tex]:
[tex]\[ h^2 + 6h + 5 = (h + 1)(h + 5) \][/tex]
The factors are [tex]\((h + 1)\)[/tex] and [tex]\((h + 5)\)[/tex]. No factor of [tex]\((h + 2)\)[/tex] here.
Expression C: [tex]\(2h^2 - 4h\)[/tex]
Factor out the greatest common factor (GCF):
[tex]\[ 2h^2 - 4h = 2h(h - 2) \][/tex]
The factors are [tex]\(2h\)[/tex] and [tex]\((h - 2)\)[/tex]. No factor of [tex]\((h + 2)\)[/tex] here.
Expression D: [tex]\(h^2 + h - 2\)[/tex]
To factor the quadratic expression, we need to find two numbers that multiply to [tex]\(-2\)[/tex] and add up to [tex]\(1\)[/tex]. These numbers are [tex]\(2\)[/tex] and [tex]\(-1\)[/tex]:
[tex]\[ h^2 + h - 2 = (h + 2)(h - 1) \][/tex]
The factors are [tex]\((h + 2)\)[/tex] and [tex]\((h - 1)\)[/tex].
Since [tex]\((h + 2)\)[/tex] is a factor of Expression D, the correct answer is:
D. [tex]\(h^2 + h - 2\)[/tex]
