Let's solve each expression step-by-step.
1. Simplify [tex]\( 12 x^2 + 6 \)[/tex]:
This expression cannot be factored further or simplified to a more compact form. Therefore:
[tex]\[ 12 x^2 + 6 \][/tex]
2. Simplify [tex]\( 5 x y^3 - 10 \)[/tex]:
This expression does not contain common factors that can be extracted. Therefore:
[tex]\[ 5 x y^3 - 10 \][/tex]
3. Simplify [tex]\( m^2 n^3 - m n^2 \)[/tex]:
Factor out the common term [tex]\( m n^2 \)[/tex]:
[tex]\[ m^2 n^3 - m n^2 = m n^2 (m n - 1) \][/tex]
4. Simplify [tex]\( 16 y^2 - 8 \)[/tex]:
Factor out the common term 8:
[tex]\[ 16 y^2 - 8 = 8 (2 y^2 - 1) \][/tex]
5. Simplify [tex]\( 24 b^5 - 36 b^4 \)[/tex]:
Factor out the common term [tex]\( 12 b^4 \)[/tex]:
[tex]\[ 24 b^5 - 36 b^4 = 12 b^4 (2 b - 3) \][/tex]
6. Simplify [tex]\( a^2 b c + a b^2 c + 3 a b c^2 \)[/tex]:
Factor out the common term [tex]\( a b c \)[/tex]:
[tex]\[ a^2 b c + a b^2 c + 3 a b c^2 = a b c (a + b + 3 c) \][/tex]
7. Simplify [tex]\( 4 m^2 n^2 - 4 m n^3 \)[/tex]:
Factor out the common term [tex]\( 4 m n^2 \)[/tex]:
[tex]\[ 4 m^2 n^2 - 4 m n^3 = 4 m n^2 (m - n) \][/tex]
8. Simplify [tex]\( 25 a + 25 b \)[/tex]:
Factor out the common term 25:
[tex]\[ 25 a + 25 b = 25 (a + b) \][/tex]
9. Simplify [tex]\( 3 x^2 + 9 x y \)[/tex]:
Factor out the common term [tex]\( 3 x \)[/tex]:
[tex]\[ 3 x^2 + 9 x y = 3 x (x + 3 y) \][/tex]
10. Simplify [tex]\( 2 x^2 y + 12 x y \)[/tex]:
Factor out the common term [tex]\( 2 x y \)[/tex]:
[tex]\[ 2 x^2 y + 12 x y = 2 x y (x + 6) \][/tex]
So, the final simplified forms are:
1. [tex]\( 12 x^2 + 6 \)[/tex]
2. [tex]\( 5 x y^3 - 10 \)[/tex]
3. [tex]\( m n^2 (m n - 1) \)[/tex]
4. [tex]\( 16 y^2 - 8 \)[/tex]
5. [tex]\( b^4 (24 b - 36) \)[/tex]
6. [tex]\( a b c (a + b + 3 c) \)[/tex]
7. [tex]\( 4 m n^2 (m - n) \)[/tex]
8. [tex]\( 25 (a + b) \)[/tex]
9. [tex]\( 3 x (x + 3 y) \)[/tex]
10. [tex]\( 2 x y (x + 6) \)[/tex]
