Question

question 1-4
match each polynomial with a factoring technique. if none of the techniques can be used to factor the polynomial, select none.

	factor by grouping	difference of squares	perfect square trinomial	none
$16m^2 + 40mn + 25n^2$	$\square$	$\square$	$\square$	$\square$
$64d^2 + 9$	$\square$	$\square$	$\square$	$\square$
$9a^2 - 36b^2$	$\square$	$\square$	$\square$	$\square$

Explanation:

For \( 8xy + 4y + 10x + 5 \)

Step1: Group terms

Group as \( (8xy + 4y) + (10x + 5) \)

Step2: Factor each group

Factor \( 4y \) from first group: \( 4y(2x + 1) \), factor \( 5 \) from second group: \( 5(2x + 1) \)

Step3: Factor out common binomial

Factor out \( (2x + 1) \): \( (4y + 5)(2x + 1) \), so factor by grouping applies.

Step1: Check perfect square trinomial form

A perfect square trinomial is \( (a + b)^2 = a^2 + 2ab + b^2 \). Here, \( a = 4m \), \( b = 5n \), \( 2ab = 2(4m)(5n)=40mn \), \( a^2 = 16m^2 \), \( b^2 = 25n^2 \), so it's \( (4m + 5n)^2 \), so perfect square trinomial applies.

Step1: Check difference of squares (needs \( a^2 - b^2 \)), but this is \( (8d)^2 + 3^2 \), sum of squares (not factorable over reals with these techniques), and not groupable or perfect square trinomial. So none applies.

Answer:

Factor by grouping (check the box for \( 8xy + 4y + 10x + 5 \) under "Factor by grouping")

For \( 16m^2 + 40mn + 25n^2 \)