Carry out the addition and simplify by collecting like terms: (a) (𝑥2 + 2𝑥 + 1) + (3𝑥 + 7) (b) (𝑛2 + 2𝑚) + (𝑚 − 𝑛2) (c) (𝑦5 − 2𝑦3 + 𝑦) + (2𝑦5 + 𝑦3 − 𝑦) (d) (𝑥2𝑦 − 7𝑦) + (𝑦 + 7𝑥2𝑦) 3) 4) (e) (𝑥𝑦 − 1) + (−2𝑥𝑦 + 5)

1. Align like terms: x^2 (no matching term), 2x + 3x , 1 + 7 .
2. Simplify: x^2 , 5x , 8 .
3. Result: x^2 + 5x + 8 .

(b) \left(n^2 + 2m\right) + \left(m - n^2\right)

= n^2 + 2m + m - n^2

[Solution] 3m

[Step-by-Step]
1. Align like terms: n^2 - n^2 , 2m + m .
2. Simplify: 0 , 3m .
3. Result: 3m .

(c) \left(y^5 - 2y^3 + y\right) + \left(2y^5 + y^3 - y\right)

= y^5 - 2y^3 + y + 2y^5 + y^3 - y

[Solution] 3y^5 - y^3

[Step-by-Step]
1. Align like terms: y^5 + 2y^5 , -2y^3 + y^3 , y - y .
2. Simplify: 3y^5 , -y^3 , 0 .
3. Result: 3y^5 - y^3 .

(d) \left(x^2y - 7y\right) + \left(y + 7x^2y\right)

= x^2y - 7y + y + 7x^2y

[Solution] 8x^2y - 6y

[Step-by-Step]
1. Align like terms: x^2y + 7x^2y , -7y + y .
2. Simplify: 8x^2y , -6y .
3. Result: 8x^2y - 6y .

(e) \left(xy - 1\right) + \left(-2xy + 5\right)

= xy - 1 - 2xy + 5

[Solution] -xy + 4

[Step-by-Step]
1. Align like terms: xy - 2xy , -1 + 5 .
2. Simplify: -xy , 4 .
3. Result: -xy + 4 .