Question

faelyn grouped the terms and factored the gcf out of the groups of the polynomial $6x^4 - 8x^3 + 3x^2 + 4$. her work is shown. step 1: $(6x^4 - 8x^3) + (3x^2 + 4)$ step 2: $2x^2(3x^2 - 4) + 1(3x^2 + 4)$ faelyn noticed that she does not have a common factor. which accurately describes what faelyn should do next? \bigcirc faelyn should realize that her work shows that the polynomial is prime. \bigcirc faelyn should go back and regroup the terms in step 1 as $(6x^4 + 3x^2) - (8x^3 + 4)$. \bigcirc in step 2, faelyn should factor only $2x$ out of the first expression. \bigcirc faelyn should factor out a negative from one of the groups so the binomials will be the same.

Explanation:

To factor by grouping, after factoring GCF from groups, the binomials inside should be the same to factor further. Here, current groups have \(3x^2 - 4\) and \(3x^2 + 4\) (not same). If we regroup as \((6x^4 + 3x^2)-(8x^2 + 4)\), factor GCF: \(3x^2(2x^2 + 1)-4(2x^2 + 1)\), then common binomial \(2x^2 + 1\) can be factored. Other options: First is wrong (polynomial is factorable by regrouping), third is incorrect factoring, fourth - factoring negative won't make binomials same here. So regrouping terms in Step 1 as \((6x^4 + 3x^2)-(8x^2 + 4)\) is correct.

Answer:

B. Faelyn should go back and regroup the terms in Step 1 as \((6x^4 + 3x^2) - (8x^2 + 4)\).