Question

1. $3r^2 + 40r + 100$
2. $5p^2 + 19p + 12$
3. $3n^2 + 10n - 8$
4. $10n^2 - 21n - 49$
5. $9x^2 + 9x - 40$
6. $4m^2 - 4m - 63$
7. $4x^2 - 35x + 24$
8. $6k^2 - 10k + 50$

Explanation:

Let's solve one of them, say problem 15: \(3r^2 + 40r + 100\)

Step1: Find two numbers

We need two numbers that multiply to \(3\times100 = 300\) and add up to \(40\). The numbers are \(30\) and \(10\) since \(30\times10 = 300\) and \(30 + 10 = 40\).

Step2: Split the middle term

Rewrite the middle term using these numbers: \(3r^2 + 30r + 10r + 100\)

Step3: Factor by grouping

Group the first two and last two terms: \((3r^2 + 30r) + (10r + 100)\)
Factor out the GCF from each group: \(3r(r + 10) + 10(r + 10)\)

Step4: Factor out the common binomial

Factor out \((r + 10)\): \((3r + 10)(r + 10)\)

Now problem 17: \(5p^2 + 19p + 12\)

Step1: Find two numbers

Multiply \(5\times12 = 60\), find two numbers that add to \(19\) and multiply to \(60\). The numbers are \(15\) and \(4\) ( \(15\times4 = 60\), \(15 + 4 = 19\) )

Step2: Split the middle term

\(5p^2 + 15p + 4p + 12\)

Step3: Factor by grouping

\((5p^2 + 15p) + (4p + 12)\)
Factor out GCF: \(5p(p + 3) + 4(p + 3)\)

Step4: Factor out common binomial

\((5p + 4)(p + 3)\)

Problem 19: \(3n^2 + 10n - 8\)

Step1: Find two numbers

Multiply \(3\times(-8) = -24\), find two numbers that add to \(10\) and multiply to \(-24\). The numbers are \(12\) and \(-2\) ( \(12\times(-2) = -24\), \(12 + (-2) = 10\) )

Step2: Split the middle term

\(3n^2 + 12n - 2n - 8\)

Step3: Factor by grouping

\((3n^2 + 12n) + (-2n - 8)\)
Factor out GCF: \(3n(n + 4) - 2(n + 4)\)

Step4: Factor out common binomial

\((3n - 2)(n + 4)\)

Problem 21: \(10n^2 - 21n - 49\)

Step1: Find two numbers

Multiply \(10\times(-49) = -490\), find two numbers that add to \(-21\) and multiply to \(-490\). The numbers are \(-35\) and \(14\) ( \(-35\times14 = -490\), \(-35 + 14 = -21\) )

Step2: Split the middle term

\(10n^2 - 35n + 14n - 49\)

Step3: Factor by grouping

\((10n^2 - 35n) + (14n - 49)\)
Factor out GCF: \(5n(2n - 7) + 7(2n - 7)\)

Step4: Factor out common binomial

\((5n + 7)(2n - 7)\)

Problem 23: \(9x^2 + 9x - 40\)

Step1: Find two numbers

Multiply \(9\times(-40) = -360\), find two numbers that add to \(9\) and multiply to \(-360\). The numbers are \(24\) and \(-15\) ( \(24\times(-15) = -360\), \(24 + (-15) = 9\) )

Step2: Split the middle term

\(9x^2 + 24x - 15x - 40\)

Step3: Factor by grouping

\((9x^2 + 24x) + (-15x - 40)\)
Factor out GCF: \(3x(3x + 8) - 5(3x + 8)\)

Step4: Factor out common binomial

\((3x - 5)(3x + 8)\)

Problem 25: \(4m^2 - 4m - 63\)

Step1: Find two numbers

Multiply \(4\times(-63) = -252\), find two numbers that add to \(-4\) and multiply to \(-252\). The numbers are \(14\) and \(-18\) ( \(14\times(-18) = -252\), \(14 + (-18) = -4\) )

Step2: Split the middle term

\(4m^2 + 14m - 18m - 63\)

Step3: Factor by grouping

\((4m^2 + 14m) + (-18m - 63)\)
Factor out GCF: \(2m(2m + 7) - 9(2m + 7)\)

Step4: Factor out common binomial

\((2m - 9)(2m + 7)\)

Problem 27: \(4x^2 - 35x + 24\)

Step1: Find two numbers

Multiply \(4\times24 = 96\), find two numbers that add to \(-35\) and multiply to \(96\). The numbers are \(-32\) and \(-3\) ( \(-32\times(-3) = 96\), \(-32 + (-3) = -35\) )

Step2: Split the middle term

\(4x^2 - 32x - 3x + 24\)

Step3: Factor by grouping

\((4x^2 - 32x) + (-3x + 24)\)
Factor out GCF: \(4x(x - 8) - 3(x - 8)\)

Step4: Factor out common binomial

\((4x - 3)(x - 8)\)

Problem 29: \(6k^2 - 10k + 50\)

Step1: Factor out GCF

The GCF of \(6\), \(-10\), and \(50\) is \(2\). Factor out \(2\): \(2(3k^2 - 5k + 25)\)
(Note: \(3k^2 - 5k + 25\) does not factor further over the real numbers as the discriminant \(b^2 - 4ac = (-5)^2 - 4\times3\times25 = 25…

Answer:

1. \((3r + 10)(r + 10)\)
2. \((5p + 4)(p + 3)\)
3. \((3n - 2)(n + 4)\)
4. \((5n + 7)(2n - 7)\)
5. \((3x - 5)(3x + 8)\)
6. \((2m - 9)(2m + 7)\)
7. \((4x - 3)(x - 8)\)
8. \(2(3k^2 - 5k + 25)\)