Sure, let's simplify each expression step by step.
(a) [tex]\(-2(x + y)\)[/tex]
To simplify this, we apply the distributive property. The distributive property states that [tex]\( a(b + c) = ab + ac \)[/tex].
[tex]\[
-2(x + y) = -2 \cdot x + (-2) \cdot y = -2x - 2y
\][/tex]
So, the simplified form is:
[tex]\[
-2x - 2y
\][/tex]
(d) [tex]\(-x(-x - y)\)[/tex]
Again, we use the distributive property.
[tex]\[
-x(-x - y) = -x \cdot (-x) + (-x) \cdot (-y) = x^2 + xy
\][/tex]
So, the simplified form is:
[tex]\[
x^2 + xy
\][/tex]
(g) [tex]\(-2x(-x + 4y)\)[/tex]
Using the distributive property:
[tex]\[
-2x(-x + 4y) = -2x \cdot (-x) + (-2x) \cdot 4y = 2x^2 - 8xy
\][/tex]
So, the simplified form is:
[tex]\[
2x^2 - 8xy
\][/tex]
(i) [tex]\(-x^4(x^2 - x)\)[/tex]
Distribute [tex]\(-x^4\)[/tex]:
[tex]\[
-x^4(x^2 - x) = -x^4 \cdot x^2 + (-x^4) \cdot (-x) = -x^6 + x^5
\][/tex]
So, the simplified form is:
[tex]\[
-x^6 + x^5
\][/tex]
(m) [tex]\(-5x(-2x^2 + 3x - 1)\)[/tex]
Applying the distributive property:
[tex]\[
-5x(-2x^2 + 3x - 1) = -5x \cdot (-2x^2) + (-5x) \cdot 3x + (-5x) \cdot (-1)
\][/tex]
Calculate each term:
[tex]\[
= 10x^3 - 15x^2 + 5x
\][/tex]
So, the simplified form is:
[tex]\[
10x^3 - 15x^2 + 5x
\][/tex]
To summarize:
(a) [tex]\(-2(x + y) = -2x - 2y\)[/tex]
(d) [tex]\(-x(-x - y) = x^2 + xy\)[/tex]
(g) [tex]\(-2x(-x + 4y) = 2x^2 - 8xy\)[/tex]
(i) [tex]\(-x^4(x^2 - x) = -x^6 + x^5\)[/tex]
(m) [tex]\(-5x(-2x^2 + 3x - 1) = 10x^3 - 15x^2 + 5x\)[/tex]
