Certainly! Let's expand and simplify each of the given expressions step by step:
### a) [tex]\( 5\left(4 x^2-2 x+3\right) \)[/tex]
First, distribute the 5 to each term inside the parentheses:
[tex]\[ 5 \cdot 4 x^2 + 5 \cdot (-2 x) + 5 \cdot 3 \][/tex]
[tex]\[ = 20 x^2 - 10 x + 15 \][/tex]
So, the simplified expression is:
[tex]\[ 20 x^2 - 10 x + 15 \][/tex]
### b) [tex]\( -4 x^2+\left(-2 x^2\right) \)[/tex]
Combine the like terms:
[tex]\[ -4 x^2 + (-2 x^2) \][/tex]
[tex]\[ = -4 x^2 - 2 x^2 \][/tex]
[tex]\[ = -6 x^2 \][/tex]
So, the simplified expression is:
[tex]\[ -6 x^2 \][/tex]
### c) [tex]\( 2\left(x^2+4 x-5\right)+\left(6+x^2\right) \)[/tex]
First, distribute the 2 to each term inside the first set of parentheses:
[tex]\[ 2 \cdot x^2 + 2 \cdot 4 x + 2 \cdot (-5) = 2 x^2 + 8 x - 10 \][/tex]
Now, add the expression inside the second set of parentheses:
[tex]\[ 2 x^2 + 8 x - 10 + 6 + x^2 \][/tex]
Combine the like terms:
[tex]\[ 2 x^2 + x^2 + 8 x + (-10 + 6) \][/tex]
[tex]\[ = 3 x^2 + 8 x - 4 \][/tex]
So, the simplified expression is:
[tex]\[ 3 x^2 + 8 x - 4 \][/tex]
### d) [tex]\( 3 x^2+4 x-x^2-x \)[/tex]
Combine the like terms:
[tex]\[ 3 x^2 - x^2 + 4 x - x \][/tex]
[tex]\[ = (3 - 1) x^2 + (4 - 1) x \][/tex]
[tex]\[ = 2 x^2 + 3 x \][/tex]
So, the simplified expression is:
[tex]\[ 2 x^2 + 3 x \][/tex]
### e) [tex]\( 7 x^3-3 x+6 x^3-x \)[/tex]
Combine the like terms:
[tex]\[ 7 x^3 + 6 x^3 - 3 x - x \][/tex]
[tex]\[ = (7 + 6) x^3 + (-3 - 1) x \][/tex]
[tex]\[ = 13 x^3 - 4 x \][/tex]
So, the simplified expression is:
[tex]\[ 13 x^3 - 4 x \][/tex]
In summary, the expanded and simplified forms of the expressions are:
a) [tex]\( 20 x^2 - 10 x + 15 \)[/tex]
b) [tex]\( -6 x^2 \)[/tex]
c) [tex]\( 3 x^2 + 8 x - 4 \)[/tex]
d) [tex]\( 2 x^2 + 3 x \)[/tex]
e) [tex]\( 13 x^3 - 4 x \)[/tex]
