6. Vicky factored 30x2 + 76x + 48 as follows:

Line 1 30x2 + 76x + 48
Line 2 30x2 + 36x + 40x + 48
Line 3 6x(5x + 6) + 8(5x +6)
Line 4 (6x + 8)(5x + 6)
A. factored Did Vicky completely factor the polynomial? Justify your response by explaining why it is is completely or by showing the correct solution if it is not factored completely.
B. Would the problem produce a different result if Line 2 was written as 30x2 + 40x + 36x + 48? Explain why or why not.

B.

line 2:
30x^2+40x + 36x+ 48
10x(3x+4) + 12(3x+4)

(10x+12)(3x+4)
2(5x+6)(3x+4)

now answer a).

A. Vicky did completely factor the polynomial. In Line 4, she factored out the greatest common factor of both terms, which is 2, and she also factored the binomial (5x + 6) that resulted from line 3. Thus, the polynomial is completely factored as (6x + 8)(5x + 6).

B. If Line 2 was written as 30x^2 + 40x + 36x + 48, the result would be different. In Line 3, Vicky grouped the terms as 6x(5x + 6) + 8(5x + 6), which allows her to factor out the common binomial term of (5x + 6). If the terms were written as 30x^2 + 40x + 36x + 48, the grouping would be different, and Vicky would not be able to factor out the binomial term correctly. The correct grouping in Line 2 is 30x^2 + 36x + 40x + 48 in order to correctly factor the binomial term in Line 3.