Question

1. 7n(3n + 1)
2. 4(2m + 3)-4(3m - 2)
3. (2n + 1)^2
4. (3x - 2)^2

Explanation:

1. For \(7n(3n + 1)\):

• Explanation:
• Step1: Use distributive property
• \(7n(3n + 1)=7n\times3n+7n\times1\).
• Step2: Multiply coefficients and variables
• \(7n\times3n = 21n^{2}\) and \(7n\times1=7n\). So \(7n(3n + 1)=21n^{2}+7n\).

1. For \(4(2m + 3)-4(3m - 1)\):

• Explanation:
• Step1: Apply distributive property to both terms
• \(4(2m + 3)=4\times2m+4\times3 = 8m + 12\) and \(4(3m - 1)=4\times3m-4\times1=12m - 4\).
• Step2: Subtract the two - expanded expressions
• \((8m + 12)-(12m - 4)=8m + 12-12m + 4\).
• Step3: Combine like - terms
• \((8m-12m)+(12 + 4)=-4m+16\).

1. For \((2n + 1)^{2}\):

• Explanation:
• Step1: Use the formula \((a + b)^{2}=a^{2}+2ab + b^{2}\), where \(a = 2n\) and \(b = 1\)
• \((2n + 1)^{2}=(2n)^{2}+2\times(2n)\times1+1^{2}\).
• Step2: Simplify each term
• \((2n)^{2}=4n^{2}\), \(2\times(2n)\times1 = 4n\), and \(1^{2}=1\). So \((2n + 1)^{2}=4n^{2}+4n + 1\).

1. For \((3x - 2)^{2}\):

• Explanation:
• Step1: Use the formula \((a - b)^{2}=a^{2}-2ab + b^{2}\), where \(a = 3x\) and \(b = 2\)
• \((3x - 2)^{2}=(3x)^{2}-2\times(3x)\times2+2^{2}\).
• Step2: Simplify each term
• \((3x)^{2}=9x^{2}\), \(2\times(3x)\times2 = 12x\), and \(2^{2}=4\). So \((3x - 2)^{2}=9x^{2}-12x + 4\).

Answer:

1. \(21n^{2}+7n\)
2. \(-4m + 16\)
3. \(4n^{2}+4n + 1\)
4. \(9x^{2}-12x + 4\)