Sure, let's go through the factoring step by step for each expression:
### 7) [tex]\( 4x^2 - 16xy + 16y^2 \)[/tex]
To factor the expression, notice that it resembles a perfect square trinomial:
[tex]\[ 4x^2 - 16xy + 16y^2 = (2x - 4y)^2 \][/tex]
Further simplifying, we get:
[tex]\[ 4(x - 2y)^2 \][/tex]
So, the factored form is:
[tex]\[ 4(x - 2y)^2 \][/tex]
### 8) [tex]\( 8x^3 - 64y^3 \)[/tex]
This expression can be factored as the difference of cubes:
[tex]\[ 8x^3 - 64y^3 = (2x)^3 - (4y)^3 \][/tex]
Using the difference of cubes formula [tex]\( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)[/tex], where [tex]\( a = 2x \)[/tex] and [tex]\( b = 4y \)[/tex], we get:
[tex]\[ (2x - 4y)((2x)^2 + (2x)(4y) + (4y)^2) \][/tex]
Simplifying the terms inside the parentheses:
[tex]\[ (2x - 4y)(4x^2 + 8xy + 16y^2) \][/tex]
So, the factored form is:
[tex]\[ 8(x - 2y)(x^2 + 2xy + 4y^2) \][/tex]
### 9) [tex]\( 5x^2 + 8x - 4 \)[/tex]
To factor this quadratic expression, we look for two numbers that multiply to [tex]\( 5(-4) = -20 \)[/tex] and add to [tex]\( 8 \)[/tex]. These numbers are [tex]\( 10 \)[/tex] and [tex]\( -2 \)[/tex].
We split the middle term using these numbers:
[tex]\[ 5x^2 + 10x - 2x - 4 \][/tex]
Then, we group the terms and factor by grouping:
[tex]\[ 5x(x + 2) - 2(x + 2) \][/tex]
Factoring out the common term [tex]\( (x + 2) \)[/tex], we get:
[tex]\[ (x + 2)(5x - 2) \][/tex]
So, the factored form is:
[tex]\[ (x + 2)(5x - 2) \][/tex]
### 10) [tex]\( 5x^3 + 3xy + 6y^2 + 10x^2 y \)[/tex]
To factor this expression, we first look for common factors in the terms:
Group terms in pairs:
[tex]\[ (5x^3 + 10x^2y) + (3xy + 6y^2) \][/tex]
From the first group, we can factor out [tex]\( 5x^2 \)[/tex]:
[tex]\[ 5x^2(x + 2y) \][/tex]
From the second group, we can factor out [tex]\( 3y \)[/tex]:
[tex]\[ 3y(x + 2y) \][/tex]
Combining these groups, we get:
[tex]\[ 5x^2(x + 2y) + 3y(x + 2y) \][/tex]
We can factor out the common binomial [tex]\( (x + 2y) \)[/tex]:
[tex]\[ (x + 2y)(5x^2 + 3y) \][/tex]
So, the factored form is:
[tex]\[ (x + 2y)(5x^2 + 3y) \][/tex]
In summary, the factored forms of the given expressions are:
7) [tex]\( 4(x - 2y)^2 \)[/tex]
8) [tex]\( 8(x - 2y)(x^2 + 2xy + 4y^2) \)[/tex]
9) [tex]\( (x + 2)(5x - 2) \)[/tex]
10) [tex]\( (x + 2y)(5x^2 + 3y) \)[/tex]
