Answer :
Let's simplify each of the given expressions step-by-step:
1) Simplify [tex]\( x^2 + 4x - 5x + x^2 \)[/tex]:
First, combine the like terms. Here, [tex]\( x^2 \)[/tex] terms and the [tex]\( x \)[/tex] terms are like terms.
[tex]\[ x^2 + x^2 + 4x - 5x \][/tex]
Combine the [tex]\( x^2 \)[/tex] terms:
[tex]\[ (x^2 + x^2) = 2x^2 \][/tex]
Combine the [tex]\( x \)[/tex] terms:
[tex]\[ 4x - 5x = - x \][/tex]
So, the expression simplifies to:
[tex]\[ 2x^2 - x \][/tex]
This can also be factored as:
[tex]\[ x(2x - 1) \][/tex]
Thus,
[tex]\[ x^2 + 4x - 5x + x^2 = \boxed{x(2x - 1)} \][/tex]
2) Simplify [tex]\( 5a^4 - 3a^2 + a^4 + 4a^2 \)[/tex]:
First, combine the [tex]\( a^4 \)[/tex] terms and the [tex]\( a^2 \)[/tex] terms.
[tex]\[ 5a^4 + a^4 - 3a^2 + 4a^2 \][/tex]
Combine the [tex]\( a^4 \)[/tex] terms:
[tex]\[ 5a^4 + a^4 = 6a^4 \][/tex]
Combine the [tex]\( a^2 \)[/tex] terms:
[tex]\[ -3a^2 + 4a^2 = a^2 \][/tex]
So, the expression simplifies to:
[tex]\[ 6a^4 + a^2 \][/tex]
Thus,
[tex]\[ 5a^4 - 3a^2 + a^4 + 4a^2 = \boxed{6a^4 + a^2} \][/tex]
3) Simplify [tex]\( x^3 - 2x - x + 4 \)[/tex]:
First, combine the [tex]\( x \)[/tex] terms:
[tex]\[ x^3 - 2x - x + 4 \][/tex]
Combine the [tex]\( x \)[/tex] terms:
[tex]\[ -2x - x = -3x \][/tex]
So, the expression simplifies to:
[tex]\[ x^3 - 3x + 4 \][/tex]
Thus,
[tex]\[ x^3 - 2x - x + 4 = \boxed{x^3 - 3x + 4} \][/tex]
4) Simplify [tex]\( x^2 - 4x - 7x + 6 + 3x^2 - 5 \)[/tex]:
First, combine the [tex]\( x^2 \)[/tex] terms, the [tex]\( x \)[/tex] terms, and the constants:
[tex]\[ x^2 + 3x^2 - 4x - 7x + 6 - 5 \][/tex]
Combine the [tex]\( x^2 \)[/tex] terms:
[tex]\[ x^2 + 3x^2 = 4x^2 \][/tex]
Combine the [tex]\( x \)[/tex] terms:
[tex]\[ -4x - 7x = -11x \][/tex]
Combine the constants:
[tex]\[ 6 - 5 = 1 \][/tex]
So, the expression simplifies to:
[tex]\[ 4x^2 - 11x + 1 \][/tex]
Thus,
[tex]\[ x^2 - 4x - 7x + 6 + 3x^2 - 5 = \boxed{4x^2 - 11x + 1} \][/tex]
5) Simplify [tex]\( m^2 + n^2 - 3n + 4n^2 - 5m^2 - 5n^2 \)[/tex]:
Combine the [tex]\( m^2 \)[/tex] terms and the [tex]\( n^2 \)[/tex] terms, and consider the constants separately:
[tex]\[ m^2 - 5m^2 + n^2 + 4n^2 - 5n^2 - 3n \][/tex]
Combine the [tex]\( m^2 \)[/tex] terms:
[tex]\[ m^2 - 5m^2 = -4m^2 \][/tex]
Combine the [tex]\( n^2 \)[/tex] terms:
[tex]\[ n^2 + 4n^2 - 5n^2 = 0 \][/tex]
Combine the [tex]\( n \)[/tex] terms:
[tex]\[ -3n \][/tex]
So, the expression simplifies to:
[tex]\[ -4m^2 - 3n \][/tex]
Thus,
[tex]\[ m^2 + n^2 - 3n + 4n^2 - 5m^2 - 5n^2 = \boxed{-4m^2 - 3n} \][/tex]
6) Simplify [tex]\( 3x - x^3 - 4x + 5 + x^3 + 4x^2 - 6 \)[/tex]:
First, combine the [tex]\( x^3 \)[/tex] terms, the [tex]\( x^2 \)[/tex] terms, the [tex]\( x \)[/tex] terms, and the constants:
[tex]\[ - x^3 + x^3 + 4x^2 + 3x - 4x + 5 - 6 \][/tex]
Combine the [tex]\( x^3 \)[/tex] terms:
[tex]\[ -x^3 + x^3 = 0 \][/tex]
Combine the [tex]\( x^2 \)[/tex] terms:
[tex]\[ 4x^2 \][/tex]
Combine the [tex]\( x \)[/tex] terms:
[tex]\[ 3x - 4x = -x \][/tex]
Combine the constants:
[tex]\[ 5 - 6 = -1 \][/tex]
So, the expression simplifies to:
[tex]\[ 4x^2 - x - 1 \][/tex]
Thus,
[tex]\[ 3x - x^3 - 4x + 5 + x^3 + 4x^2 - 6 = \boxed{4x^2 - x - 1} \][/tex]
1) Simplify [tex]\( x^2 + 4x - 5x + x^2 \)[/tex]:
First, combine the like terms. Here, [tex]\( x^2 \)[/tex] terms and the [tex]\( x \)[/tex] terms are like terms.
[tex]\[ x^2 + x^2 + 4x - 5x \][/tex]
Combine the [tex]\( x^2 \)[/tex] terms:
[tex]\[ (x^2 + x^2) = 2x^2 \][/tex]
Combine the [tex]\( x \)[/tex] terms:
[tex]\[ 4x - 5x = - x \][/tex]
So, the expression simplifies to:
[tex]\[ 2x^2 - x \][/tex]
This can also be factored as:
[tex]\[ x(2x - 1) \][/tex]
Thus,
[tex]\[ x^2 + 4x - 5x + x^2 = \boxed{x(2x - 1)} \][/tex]
2) Simplify [tex]\( 5a^4 - 3a^2 + a^4 + 4a^2 \)[/tex]:
First, combine the [tex]\( a^4 \)[/tex] terms and the [tex]\( a^2 \)[/tex] terms.
[tex]\[ 5a^4 + a^4 - 3a^2 + 4a^2 \][/tex]
Combine the [tex]\( a^4 \)[/tex] terms:
[tex]\[ 5a^4 + a^4 = 6a^4 \][/tex]
Combine the [tex]\( a^2 \)[/tex] terms:
[tex]\[ -3a^2 + 4a^2 = a^2 \][/tex]
So, the expression simplifies to:
[tex]\[ 6a^4 + a^2 \][/tex]
Thus,
[tex]\[ 5a^4 - 3a^2 + a^4 + 4a^2 = \boxed{6a^4 + a^2} \][/tex]
3) Simplify [tex]\( x^3 - 2x - x + 4 \)[/tex]:
First, combine the [tex]\( x \)[/tex] terms:
[tex]\[ x^3 - 2x - x + 4 \][/tex]
Combine the [tex]\( x \)[/tex] terms:
[tex]\[ -2x - x = -3x \][/tex]
So, the expression simplifies to:
[tex]\[ x^3 - 3x + 4 \][/tex]
Thus,
[tex]\[ x^3 - 2x - x + 4 = \boxed{x^3 - 3x + 4} \][/tex]
4) Simplify [tex]\( x^2 - 4x - 7x + 6 + 3x^2 - 5 \)[/tex]:
First, combine the [tex]\( x^2 \)[/tex] terms, the [tex]\( x \)[/tex] terms, and the constants:
[tex]\[ x^2 + 3x^2 - 4x - 7x + 6 - 5 \][/tex]
Combine the [tex]\( x^2 \)[/tex] terms:
[tex]\[ x^2 + 3x^2 = 4x^2 \][/tex]
Combine the [tex]\( x \)[/tex] terms:
[tex]\[ -4x - 7x = -11x \][/tex]
Combine the constants:
[tex]\[ 6 - 5 = 1 \][/tex]
So, the expression simplifies to:
[tex]\[ 4x^2 - 11x + 1 \][/tex]
Thus,
[tex]\[ x^2 - 4x - 7x + 6 + 3x^2 - 5 = \boxed{4x^2 - 11x + 1} \][/tex]
5) Simplify [tex]\( m^2 + n^2 - 3n + 4n^2 - 5m^2 - 5n^2 \)[/tex]:
Combine the [tex]\( m^2 \)[/tex] terms and the [tex]\( n^2 \)[/tex] terms, and consider the constants separately:
[tex]\[ m^2 - 5m^2 + n^2 + 4n^2 - 5n^2 - 3n \][/tex]
Combine the [tex]\( m^2 \)[/tex] terms:
[tex]\[ m^2 - 5m^2 = -4m^2 \][/tex]
Combine the [tex]\( n^2 \)[/tex] terms:
[tex]\[ n^2 + 4n^2 - 5n^2 = 0 \][/tex]
Combine the [tex]\( n \)[/tex] terms:
[tex]\[ -3n \][/tex]
So, the expression simplifies to:
[tex]\[ -4m^2 - 3n \][/tex]
Thus,
[tex]\[ m^2 + n^2 - 3n + 4n^2 - 5m^2 - 5n^2 = \boxed{-4m^2 - 3n} \][/tex]
6) Simplify [tex]\( 3x - x^3 - 4x + 5 + x^3 + 4x^2 - 6 \)[/tex]:
First, combine the [tex]\( x^3 \)[/tex] terms, the [tex]\( x^2 \)[/tex] terms, the [tex]\( x \)[/tex] terms, and the constants:
[tex]\[ - x^3 + x^3 + 4x^2 + 3x - 4x + 5 - 6 \][/tex]
Combine the [tex]\( x^3 \)[/tex] terms:
[tex]\[ -x^3 + x^3 = 0 \][/tex]
Combine the [tex]\( x^2 \)[/tex] terms:
[tex]\[ 4x^2 \][/tex]
Combine the [tex]\( x \)[/tex] terms:
[tex]\[ 3x - 4x = -x \][/tex]
Combine the constants:
[tex]\[ 5 - 6 = -1 \][/tex]
So, the expression simplifies to:
[tex]\[ 4x^2 - x - 1 \][/tex]
Thus,
[tex]\[ 3x - x^3 - 4x + 5 + x^3 + 4x^2 - 6 = \boxed{4x^2 - x - 1} \][/tex]
