Question

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factor using any applicable method.
e1) ( b^2 + 18b + 15 )
e2) ( 12v^2 - 3v )
e3) ( 4x^2 + 16 )
e4) ( 5l^2 - 25l + 15 )
e5) ( t^2 - 0t - 36 )
e6) ( t^2 - 36 )
e7) ( xy + 2xy^2 + x^2 )
e8) ( k^2 + 8k - 20 )
e9) ( x^2y^2 - s^2j^2 )

Explanation:

E1: Step1: Check for factoring pairs

No integer pairs multiply to 15 and add to 18.

E1: Step2: Final form (prime quadratic)

$b^2 + 18b + 15$ (cannot be factored over integers)

E2: Step1: Factor out GCF

GCF of $12v^2$ and $3v$ is $3v$.
$12v^2 - 3v = 3v(4v - 1)$

E3: Step1: Factor out GCF

GCF of $4x^2$ and $16$ is $4$.
$4x^2 + 16 = 4(x^2 + 4)$

E4: Step1: Factor out GCF

GCF of $5L^2$, $25L$, $15$ is $5$.
$5L^2 - 25L + 15 = 5(L^2 - 5L + 3)$

E4: Step2: Check quadratic factor

No integer pairs multiply to 3 and add to -5.

E5: Step1: Simplify the expression

$t^2 - 0t - 36 = t^2 - 36$

E5: Step2: Factor difference of squares

Use $a^2 - b^2=(a-b)(a+b)$, where $a=t$, $b=6$.
$t^2 - 36=(t-6)(t+6)$

E6: Step1: Factor difference of squares

Use $a^2 - b^2=(a-b)(a+b)$, where $a=t$, $b=6$.
$t^2 - 36=(t-6)(t+6)$

E7: Step1: Factor out GCF

GCF of $xy$, $2xy^2$, $x^2$ is $x$.
$xy + 2xy^2 + x^2 = x(y + 2y^2 + x)$

E7: Step2: Rearrange terms (optional)

$x(2y^2 + x + y)$

E8: Step1: Find factor pairs

Find two integers that multiply to -20 and add to 8: 10 and -2.

E8: Step2: Factor the quadratic

$k^2 + 8k - 20=(k+10)(k-2)$

E9: Step1: Factor difference of squares

Use $a^2 - b^2=(a-b)(a+b)$, where $a=xy$, $b=sj$.
$x^2y^2 - s^2j^2=(xy - sj)(xy + sj)$

Answer:

E1) $b^2 + 18b + 15$ (prime over integers)
E2) $3v(4v - 1)$
E3) $4(x^2 + 4)$
E4) $5(L^2 - 5L + 3)$
E5) $(t-6)(t+6)$
E6) $(t-6)(t+6)$
E7) $x(2y^2 + x + y)$
E8) $(k+10)(k-2)$
E9) $(xy - sj)(xy + sj)$