Question

monday:
① $(7n^2 + 28n)(10n + 5)$

② $(45m - 72)(20m + 4v)$

Explanation:

Problem 1: \((7n^2 + 28n)(10n + 5)\)

Step 1: Factor out common terms

First, factor out the greatest common factor (GCF) from each binomial. For \(7n^2 + 28n\), the GCF of \(7n^2\) and \(28n\) is \(7n\), so we get \(7n(n + 4)\). For \(10n + 5\), the GCF of \(10n\) and \(5\) is \(5\), so we get \(5(2n + 1)\). Now the expression becomes \(7n(n + 4) \times 5(2n + 1)\).

Step 2: Multiply the constants and the binomials

Multiply the constant factors \(7n \times 5 = 35n\), and then multiply the binomials \((n + 4)(2n + 1)\) using the distributive property (FOIL method).
First, \(n \times 2n = 2n^2\), then \(n \times 1 = n\), then \(4 \times 2n = 8n\), and finally \(4 \times 1 = 4\). Combine like terms: \(2n^2 + n + 8n + 4 = 2n^2 + 9n + 4\).

Step 3: Multiply the results

Now multiply \(35n\) with \(2n^2 + 9n + 4\). Using the distributive property: \(35n \times 2n^2 = 70n^3\), \(35n \times 9n = 315n^2\), \(35n \times 4 = 140n\). So the expanded form is \(70n^3 + 315n^2 + 140n\). Alternatively, if we just expand the original expression without factoring first:
\((7n^2 + 28n)(10n + 5)=7n^2\times10n + 7n^2\times5 + 28n\times10n + 28n\times5 = 70n^3 + 35n^2 + 280n^2 + 140n = 70n^3 + 315n^2 + 140n\)

Problem 2: \((45m - 72)(20m + 4v)\)

Step 1: Factor out common terms (optional)

Factor out the GCF from \(45m - 72\), which is \(9\), so \(9(5m - 8)\). The second binomial \(20m + 4v\) has a GCF of \(4\), so \(4(5m + v)\). Now the expression is \(9(5m - 8) \times 4(5m + v)\).

Step 2: Multiply the constants and the binomials

Multiply the constants \(9 \times 4 = 36\), and then multiply the binomials \((5m - 8)(5m + v)\) using the distributive property.
\(5m \times 5m = 25m^2\), \(5m \times v = 5mv\), \(-8 \times 5m = -40m\), \(-8 \times v = -8v\). Combine like terms: \(25m^2 + 5mv - 40m - 8v\).

Step 3: Multiply the results

Multiply \(36\) with \(25m^2 + 5mv - 40m - 8v\): \(36\times25m^2 = 900m^2\), \(36\times5mv = 180mv\), \(36\times(-40m) = -1440m\), \(36\times(-8v) = -288v\). Alternatively, expanding the original expression directly:
\((45m - 72)(20m + 4v)=45m\times20m + 45m\times4v - 72\times20m - 72\times4v = 900m^2 + 180mv - 1440m - 288v\)

Answer:

s:

1. \(\boldsymbol{70n^3 + 315n^2 + 140n}\)
2. \(\boldsymbol{900m^2 + 180mv - 1440m - 288v}\)