Answer
a) 3xy(7y+2)
b) 5[(3x-2)(3x+2)]
c) (2y+2)(2y-3)
d) ($a^{2}$+5)(1+b)
e) (2x+y)($4x^{2}$-2xy+$y^{2}$)
Work Step by Step
a) There is a common factor of 3xy, so we divide each term by 3xy. $\frac{21xy^{2}}{3xy}$=7y. $\frac{6xy}{3xy}$=2
This then gives an answer of 3xy(7y+2)
b) There is a common factor of 5, so divide each term by 5. This gives us an answer of 5($9x^{2}$-4). The term is the bracket can be factorized further into (3x-2)(3x+2). This leaves a final answer of 5[(3x-2)(3x+2)]
c) $4y^{2}$-2y-6 is a trinomial and can be factorized into (2y+2)(2y-3). The options for the first term $4y^{2}$ are 2 and 2 or 4 and 1, however 4 and 1 together cannot give an answer of -2 so therefore 2 and 2 can. For -6 we have the option of 3 and 2 or 6 and 1 and with using 2 and 2, 3 and 2 will give the correct answer.
d) There are 2 sets of terms with common factors in this equation. the first being a common factor of $a^{2}$ in $a^{2}+a^{2}b$ and then next being 5 in 5+5b. This leaves $a^{2}$(1+b)+5(1+b). As (1+b) is common, this can be simplified into ($a^{2}$+5)(1+b).
e) The first step in this cubic equation is to cube root both terms. $\sqrt[3](8x^{3})$=2x and $\sqrt[3]y^{3}$=y this can be put as (2x+y). The next step is to square of both terms and multiply them and change the sign of the answer. $(2x)^{2}$=$4x^{2}$ and $y^{2}$. The multiplication gives 2xy and changing the sign gives -2xy. This gives an answer of ($4x^{2}-2xy+y^{2}$). This trinomial cannot be simplified further and thus a final answer of (2x-y)($4x^{2}-2xy+y^{2}$)
