Answer :
Let's verify each equation step by step:
### Equation 1:
[tex]\[ -2 b^3 (6 b^2 + 5) = -12 b^5 - 10 b^3 \][/tex]
Expand the left side:
[tex]\[ -2 b^3 (6 b^2 + 5) = -2 b^3 \cdot 6 b^2 + (-2 b^3 \cdot 5) = -12 b^5 - 10 b^3 \][/tex]
So the equation holds:
[tex]\[ -12 b^5 - 10 b^3 = -12 b^5 - 10 b^3 \][/tex]
Equation 1 is correct.
### Equation 2:
[tex]\[ -5 d^3 (3 d^2 + 4) = -15 d^5 - 20 d^3 \][/tex]
Expand the left side:
[tex]\[ -5 d^3 (3 d^2 + 4) = -5 d^3 \cdot 3 d^2 + (-5 d^3 \cdot 4) = -15 d^5 - 20 d^3 \][/tex]
So the equation holds:
[tex]\[ -15 d^5 - 20 d^3 = -15 d^5 - 20 d^3 \][/tex]
Equation 2 is correct.
### Equation 3:
[tex]\[ -3 y^4 (5 y^2 + 2) = -15 y^8 - 6 y^4 \][/tex]
Expand the left side:
[tex]\[ -3 y^4 (5 y^2 + 2) = -3 y^4 \cdot 5 y^2 + (-3 y^4 \cdot 2) = -15 y^6 - 6 y^4 \][/tex]
Compare it with the right side:
[tex]\[ -15 y^8 - 6 y^4 \neq -15 y^6 - 6 y^4 \][/tex]
Equation 3 is incorrect.
### Equation 4:
[tex]\[ -7 x^2 (4 x^3 + 2) = -28 x^6 - 14 x^2 \][/tex]
Expand the left side:
[tex]\[ -7 x^2 (4 x^3 + 2) = -7 x^2 \cdot 4 x^3 + (-7 x^2 \cdot 2) = -28 x^5 - 14 x^2 \][/tex]
Compare it with the right side:
[tex]\[ -28 x^6 - 14 x^2 \neq -28 x^5 - 14 x^2 \][/tex]
Equation 4 is incorrect.
### Conclusion:
The correct equations are:
1. [tex]\(-2 b^3 (6 b^2 + 5) = -12 b^5 - 10 b^3\)[/tex]
2. [tex]\(-5 d^3 (3 d^2 + 4) = -15 d^5 - 20 d^3\)[/tex]
### Equation 1:
[tex]\[ -2 b^3 (6 b^2 + 5) = -12 b^5 - 10 b^3 \][/tex]
Expand the left side:
[tex]\[ -2 b^3 (6 b^2 + 5) = -2 b^3 \cdot 6 b^2 + (-2 b^3 \cdot 5) = -12 b^5 - 10 b^3 \][/tex]
So the equation holds:
[tex]\[ -12 b^5 - 10 b^3 = -12 b^5 - 10 b^3 \][/tex]
Equation 1 is correct.
### Equation 2:
[tex]\[ -5 d^3 (3 d^2 + 4) = -15 d^5 - 20 d^3 \][/tex]
Expand the left side:
[tex]\[ -5 d^3 (3 d^2 + 4) = -5 d^3 \cdot 3 d^2 + (-5 d^3 \cdot 4) = -15 d^5 - 20 d^3 \][/tex]
So the equation holds:
[tex]\[ -15 d^5 - 20 d^3 = -15 d^5 - 20 d^3 \][/tex]
Equation 2 is correct.
### Equation 3:
[tex]\[ -3 y^4 (5 y^2 + 2) = -15 y^8 - 6 y^4 \][/tex]
Expand the left side:
[tex]\[ -3 y^4 (5 y^2 + 2) = -3 y^4 \cdot 5 y^2 + (-3 y^4 \cdot 2) = -15 y^6 - 6 y^4 \][/tex]
Compare it with the right side:
[tex]\[ -15 y^8 - 6 y^4 \neq -15 y^6 - 6 y^4 \][/tex]
Equation 3 is incorrect.
### Equation 4:
[tex]\[ -7 x^2 (4 x^3 + 2) = -28 x^6 - 14 x^2 \][/tex]
Expand the left side:
[tex]\[ -7 x^2 (4 x^3 + 2) = -7 x^2 \cdot 4 x^3 + (-7 x^2 \cdot 2) = -28 x^5 - 14 x^2 \][/tex]
Compare it with the right side:
[tex]\[ -28 x^6 - 14 x^2 \neq -28 x^5 - 14 x^2 \][/tex]
Equation 4 is incorrect.
### Conclusion:
The correct equations are:
1. [tex]\(-2 b^3 (6 b^2 + 5) = -12 b^5 - 10 b^3\)[/tex]
2. [tex]\(-5 d^3 (3 d^2 + 4) = -15 d^5 - 20 d^3\)[/tex]
