Question

the polynomial $2x^3 - 5x^2 + 4x - 10$ is split into two groups, $2x^3 + 4x$ and $-5x^2 - 10$. the gcfs of each group is then factored out.
what is the common binomial factor between the two groups after their gcfs have been factored out?
$\bigcirc\\ 2x + 5$
$\bigcirc\\ 2x - 5$
$\bigcirc\\ x^2 - 2$
$\bigcirc\\ x^2 + 2$

Explanation:

Step1: Factor first group

Factor GCF from \(2x^3 + 4x\). GCF is \(2x\), so \(2x(x^2 + 2)\).

Step2: Factor second group

Factor GCF from \(-5x^2 - 10\). GCF is \(-5\), so \(-5(x^2 + 2)\).

Step3: Identify common binomial

After factoring, both groups have \((x^2 + 2)\)? Wait, no, wait. Wait, original groups: \(2x^3 + 4x\) and \(-5x^2 -10\). Wait, maybe I made a mistake. Wait, let's re - factor. Wait, first group: \(2x^3+4x = 2x(x^2 + 2)\). Second group: \(-5x^2 - 10=-5(x^2 + 2)\). Wait, but the options are \(2x + 5\), \(2x-5\), \(x^2 - 2\), \(x^2+2\). Oh, wait, maybe I split the original polynomial wrong. Wait, the original polynomial is \(2x^3-5x^2 + 4x-10\). Maybe the correct grouping is \((2x^3-5x^2)+(4x - 10)\). Let's try that. First group: \(2x^3-5x^2=x^2(2x - 5)\). Second group: \(4x-10 = 2(2x - 5)\). Ah! There we go. So the first group: \(2x^3-5x^2=x^2(2x - 5)\), second group: \(4x - 10=2(2x - 5)\). So the common binomial factor is \(2x - 5\)? Wait, no, wait the original problem said the polynomial is split into \(2x^3 + 4x\) and \(-5x^2-10\). Wait, maybe the problem's grouping is \(2x^3+4x\) and \(-5x^2 - 10\). Let's re - check. \(2x^3+4x=2x(x^2 + 2)\), \(-5x^2-10=-5(x^2 + 2)\). But the options don't have \(x^2 + 2\) as the answer? Wait, no, the options do have \(x^2+2\)? Wait, the last option is \(x^2 + 2\). Wait, but maybe I misread the grouping. Wait, the original polynomial is \(2x^3-5x^2 + 4x-10\). If we group as \((2x^3 + 4x)+(-5x^2-10)\), then factoring gives \(2x(x^2 + 2)-5(x^2 + 2)=(2x - 5)(x^2 + 2)\). Wait, so the common binomial factor after factoring out GCFs is \(x^2 + 2\)? Wait, no, when we factor \(2x(x^2 + 2)-5(x^2 + 2)\), the common binomial is \(x^2 + 2\), and the other factor is \(2x-5\). But the question is "What is the common binomial factor between the two groups after their GCFs have been factored out?". So after factoring GCF from each group, the two groups are \(2x(x^2 + 2)\) and \(-5(x^2 + 2)\). So the common binomial factor is \(x^2 + 2\)? Wait, but let's check the options. The last option is \(x^2 + 2\). Wait, but maybe I made a mistake in grouping. Wait, maybe the intended grouping was \((2x^3-5x^2)+(4x - 10)\). Then first group: \(x^2(2x - 5)\), second group: \(2(2x - 5)\). Then the common binomial is \(2x - 5\), but that's not matching the first grouping. Wait, the problem says "the polynomial \(2x^3-5x^2 + 4x - 10\) is split into two groups, \(2x^3+4x\) and \(-5x^2-10\)". So we have to go with that grouping. So factoring \(2x^3 + 4x=2x(x^2 + 2)\), \(-5x^2-10=-5(x^2 + 2)\). So the common binomial factor is \(x^2 + 2\). Wait, but let's check the options. The fourth option is \(x^2 + 2\).

Answer:

\(x^2 + 2\) (the option with text " \(x^2 + 2\)")