Factoring Worksheets - Math Monks - Free Printable

Let's solve each quadratic expression by factoring. We'll go step-by-step for each problem.

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1. $ 15x^2 + 14x - 90 $

We need two numbers that:
- Multiply to $ 15 \times (-90) = -1350 $
- Add to $ 14 $

Try factor pairs of $-1350$ that add to $14$:

Try: $ 45 $ and $ -30 $:
$ 45 \times (-30) = -1350 $, $ 45 + (-30) = 15 $ → too high

Try: $ 50 $ and $ -27 $: $ 50 \times (-27) = -1350 $, $ 50 - 27 = 23 $

Try: $ 54 $ and $ -25 $: $ 54 \times (-25) = -1350 $, $ 54 - 25 = 29 $

Try: $ 60 $ and $ -22.5 $ → not integers

Wait — maybe try grouping.

Use AC method:

$ a = 15, b = 14, c = -90 $

$ ac = 15 \times (-90) = -1350 $

Find two numbers that multiply to $-1350$, add to $14$

Try: $ 45 $ and $-30$: $ 45 + (-30) = 15 $ → close

Try: $ 54 $ and $-25$: $ 54 + (-25) = 29 $

Try: $ 60 $ and $-22.5$: no

Try: $ 18 $ and $-75$: $ 18 \times (-75) = -1350 $, $ 18 - 75 = -57 $

Try: $ 30 $ and $-45$: $ 30 \times (-45) = -1350 $, $ 30 - 45 = -15 $

Try: $ 36 $ and $-37.5$: no

Wait — try $ 50 $ and $-27$: $ 50 \times (-27) = -1350 $, $ 50 - 27 = 23 $

Hmm. Try $ 60 $ and $-22.5$: not working.

Wait — perhaps factor out GCF first?

Check GCF of 15, 14, 90 → only 1.

So no common factor.

Try: $ 15x^2 + 14x - 90 $

Try factoring as $ (ax + b)(cx + d) $

Try $ (5x + a)(3x + b) $ since $ 5 \times 3 = 15 $

Try: $ (5x + 18)(3x - 5) = 15x^2 -25x + 54x -90 = 15x^2 + 29x -90 $ → too big

Try: $ (5x - 18)(3x + 5) = 15x^2 + 25x -54x -90 = 15x^2 -29x -90 $ → wrong sign

Try: $ (5x + 10)(3x - 9) = 15x^2 -45x +30x -90 = 15x^2 -15x -90 $ → no

Try: $ (5x - 10)(3x + 9) = 15x^2 +45x -30x -90 = 15x^2 +15x -90 $ → still off

Try: $ (5x + 15)(3x - 6) = 15x^2 -30x +45x -90 = 15x^2 +15x -90 $ → same

Wait — let’s use the quadratic formula to find roots and then factor.

Discriminant: $ D = 14^2 - 4(15)(-90) = 196 + 5400 = 5596 $

Not a perfect square → so it doesn’t factor nicely over integers.

Wait — maybe I made a mistake earlier.

Let me double-check: is this factorable?

Wait — let’s try again with AC method.

$ 15x^2 + 14x - 90 $

We want two numbers that multiply to $ 15 \times (-90) = -1350 $, add to $ 14 $

Try: $ 45 $ and $-30 $: sum = 15

$ 54 $ and $-25 $: sum = 29

$ 60 $ and $-22.5 $: no

$ 75 $ and $-18 $: $ 75 \times (-18) = -1350 $, $ 75 - 18 = 57 $

$ 90 $ and $-15 $: $ 90 \times (-15) = -1350 $, $ 90 - 15 = 75 $

$ 135 $ and $-10 $: $ 135 \times (-10) = -1350 $, $ 125 $

Wait — what about $ 45 $ and $-30 $? Sum is 15, too big.

Wait — try $ 30 $ and $-45 $: sum = -15

No.

Wait — $ 50 $ and $-27 $: $ 50 \times (-27) = -1350 $, $ 50 - 27 = 23 $

No.

Wait — $ 18 $ and $-75 $: $ 18 \times (-75) = -1350 $, $ 18 - 75 = -57 $

Still not.

Wait — $ 25 $ and $-54 $: $ 25 \times (-54) = -1350 $, $ 25 - 54 = -29 $

No.

Wait — $ 27 $ and $-50 $: $ 27 \times (-50) = -1350 $, $ 27 - 50 = -23 $

No.

Wait — $ 36 $ and $-37.5 $: not integer.

Wait — maybe it's not factorable? But likely it is.

Wait — try $ (15x - 18)(x + 5) = 15x^2 +75x -18x -90 = 15x^2 +57x -90 $ → no

Try $ (15x + 18)(x - 5) = 15x^2 -75x +18x -90 = 15x^2 -57x -90 $

No.

Try $ (5x + 18)(3x - 5) = 15x^2 -25x +54x -90 = 15x^2 +29x -90 $

No.

Wait — what if we try $ (3x + 10)(5x - 9) = 15x^2 -27x +50x -90 = 15x^2 +23x -90 $

Closer.

Try $ (3x - 10)(5x + 9) = 15x^2 +27x -50x -90 = 15x^2 -23x -90 $

No.

Try $ (5x - 6)(3x + 15) = 15x^2 +75x -18x -90 = 15x^2 +57x -90 $

No.

Wait — perhaps this one is not factorable with integers.

But let’s check online or recompute.

Wait — actually, let’s try using the quadratic formula:

$$
x = \frac{-14 \pm \sqrt{14^2 - 4(15)(-90)}}{2(15)} = \frac{-14 \pm \sqrt{196 + 5400}}{30} = \frac{-14 \pm \sqrt{5596}}{30}
$$

$ \sqrt{5596} \approx 75 $, $ 75^2 = 5625 $, $ 74^2 = 5476 $, $ 74.8^2 \approx 5595.04 $ → yes, ≈74.8

Not perfect square → so not factorable over integers

But that seems odd for a worksheet.

Wait — maybe I miscalculated.

Wait — let’s check if there's a typo. Is it possible it's $ 15x^2 + 14x - 90 $?

Alternatively, maybe it's supposed to be $ 15x^2 + 14x - 9 $ or something else?

Wait — let’s skip and come back.

Maybe I should try another approach.

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Let’s try #2 first.

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2. $ 16x^2 - 54x + 35 $

Use AC method:

$ a = 16, b = -54, c = 35 $

$ ac = 16 \times 35 = 560 $

Need two numbers that multiply to $ 560 $, add to $ -54 $

Try factors of 560:

- $ 40 $ and $ 14 $: $ 40 + 14 = 54 $ → so $ -40 $ and $ -14 $: $ -40 + (-14) = -54 $, $ (-40)(-14) = 560 $

Yes!

So split middle term:

$ 16x^2 - 40x - 14x + 35 $

Group:

$ (16x^2 - 40x) + (-14x + 35) $

Factor:

$ 8x(2x - 5) -7(2x - 5) $

$ = (8x - 7)(2x - 5) $

✔ Answer: $ (8x - 7)(2x - 5) $

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3. $ 24x^2 + 18x - 15 $

First, factor out GCF: all divisible by 3

$ 3(8x^2 + 6x - 5) $

Now factor $ 8x^2 + 6x - 5 $

$ ac = 8 \times (-5) = -40 $

Need two numbers that multiply to $-40$, add to $6$

Try: $ 10 $ and $-4 $: $ 10 \times (-4) = -40 $, $ 10 + (-4) = 6 $ → yes!

Split: $ 8x^2 + 10x - 4x - 5 $

Group: $ (8x^2 + 10x) + (-4x - 5) $

$ 2x(4x + 5) -1(4x + 5) = (2x - 1)(4x + 5) $

So overall: $ 3(2x - 1)(4x + 5) $

✔ Answer: $ 3(2x - 1)(4x + 5) $

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4. $ 2x^2 + 27x + 81 $

$ ac = 2 \times 81 = 162 $

Need two numbers that multiply to $162$, add to $27$

Try: $ 18 $ and $9$: $ 18 + 9 = 27 $, $ 18 \times 9 = 162 $ → yes!

Split: $ 2x^2 + 18x + 9x + 81 $

Group: $ (2x^2 + 18x) + (9x + 81) = 2x(x + 9) + 9(x + 9) = (2x + 9)(x + 9) $

✔ Answer: $ (2x + 9)(x + 9) $

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5. $ 8x^2 - 2x - 1 $

$ ac = 8 \times (-1) = -8 $

Need two numbers that multiply to $-8$, add to $-2$

Try: $ -4 $ and $2$: $ -4 + 2 = -2 $, $ -4 \times 2 = -8 $ → yes!

Split: $ 8x^2 - 4x + 2x - 1 $

Group: $ (8x^2 - 4x) + (2x - 1) = 4x(2x - 1) + 1(2x - 1) = (4x + 1)(2x - 1) $

✔ Answer: $ (4x + 1)(2x - 1) $

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6. $ 3x^2 + 6x - 9 $

GCF: 3

$ 3(x^2 + 2x - 3) $

Now factor $ x^2 + 2x - 3 $

$ (x + 3)(x - 1) $

So: $ 3(x + 3)(x - 1) $

✔ Answer: $ 3(x + 3)(x - 1) $

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7. $ 30x^2 - 27x - 21 $

GCF: 3

$ 3(10x^2 - 9x - 7) $

Now factor $ 10x^2 - 9x - 7 $

$ ac = 10 \times (-7) = -70 $

Need two numbers that multiply to $-70$, add to $-9$

Try: $ -14 $ and $5$: $ -14 + 5 = -9 $, $ -14 \times 5 = -70 $ → yes!

Split: $ 10x^2 -14x + 5x - 7 $

Group: $ (10x^2 -14x) + (5x - 7) = 2x(5x - 7) + 1(5x - 7) = (2x + 1)(5x - 7) $

So overall: $ 3(2x + 1)(5x - 7) $

✔ Answer: $ 3(2x + 1)(5x - 7) $

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8. $ 40x^2 + 126x + 44 $

GCF: 2

$ 2(20x^2 + 63x + 22) $

Now factor $ 20x^2 + 63x + 22 $

$ ac = 20 \times 22 = 440 $

Need two numbers that multiply to $440$, add to $63$

Try: $ 55 $ and $8$: $ 55 + 8 = 63 $, $ 55 \times 8 = 440 $ → yes!

Split: $ 20x^2 + 55x + 8x + 22 $

Group: $ (20x^2 + 55x) + (8x + 22) = 5x(4x + 11) + 2(4x + 11) = (5x + 2)(4x + 11) $

So overall: $ 2(5x + 2)(4x + 11) $

✔ Answer: $ 2(5x + 2)(4x + 11) $

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9. $ 5x^2 + 11x + 6 $

$ ac = 5 \times 6 = 30 $

Need two numbers that multiply to $30$, add to $11$

Try: $ 5 $ and $6$: $ 5 + 6 = 11 $, $ 5 \times 6 = 30 $ → yes!

Split: $ 5x^2 + 5x + 6x + 6 $

Group: $ (5x^2 + 5x) + (6x + 6) = 5x(x + 1) + 6(x + 1) = (5x + 6)(x + 1) $

✔ Answer: $ (5x + 6)(x + 1) $

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10. $ 7x^2 - 9x - 10 $

$ ac = 7 \times (-10) = -70 $

Need two numbers that multiply to $-70$, add to $-9$

Try: $ -14 $ and $5$: $ -14 + 5 = -9 $, $ -14 \times 5 = -70 $ → yes!

Split: $ 7x^2 -14x + 5x - 10 $

Group: $ (7x^2 -14x) + (5x - 10) = 7x(x - 2) + 5(x - 2) = (7x + 5)(x - 2) $

✔ Answer: $ (7x + 5)(x - 2) $

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Now go back to #1: $ 15x^2 + 14x - 90 $

We had trouble earlier. Let's try again.

$ ac = 15 \times (-90) = -1350 $

Need two numbers that multiply to $-1350$, add to $14$

Try: $ 45 $ and $-30 $: $ 45 + (-30) = 15 $

$ 54 $ and $-25 $: $ 54 - 25 = 29 $

$ 60 $ and $-22.5 $: no

$ 75 $ and $-18 $: $ 75 - 18 = 57 $

$ 90 $ and $-15 $: $ 90 - 15 = 75 $

$ 100 $ and $-13.5 $: no

Wait — $ 30 $ and $-45 $: $ 30 - 45 = -15 $

No.

Wait — $ 18 $ and $-75 $: $ 18 - 75 = -57 $

No.

Wait — $ 25 $ and $-54 $: $ 25 - 54 = -29 $

No.

Wait — $ 36 $ and $-37.5 $: no

Wait — $ 27 $ and $-50 $: $ 27 - 50 = -23 $

No.

Wait — $ 15 $ and $-90 $: $ 15 - 90 = -75 $

No.

Wait — $ 45 $ and $-30 $: sum 15

Close.

Wait — $ 50 $ and $-27 $: sum 23

Wait — $ 60 $ and $-22.5 $: no

Wait — maybe $ 15x^2 + 14x - 90 $ is not factorable over integers.

Let’s test discriminant:

$ D = 14^2 - 4(15)(-90) = 196 + 5400 = 5596 $

Is $ \sqrt{5596} $ rational?

$ 74^2 = 5476 $, $ 75^2 = 5625 $, so between them.

$ 74.8^2 = ? $

$ 74.8^2 = (75 - 0.2)^2 = 5625 - 2*75*0.2 + 0.04 = 5625 - 30 + 0.04 = 5595.04 $

Very close to 5596 → $ \sqrt{5596} \approx 74.81 $

Not a perfect square → so irrational roots, thus not factorable over integers

But wait — maybe I misread the problem.

Wait — could it be $ 15x^2 + 14x - 9 $ instead of -90?

Try $ 15x^2 + 14x - 9 $

Then $ ac = -135 $, need two numbers that multiply to $-135$, add to $14$

Try: $ 15 $ and $-9 $: $ 15 - 9 = 6 $

$ 27 $ and $-5 $: $ 27 - 5 = 22 $

$ 18 $ and $-7.5 $: no

$ 10 $ and $-13.5 $: no

No.

Wait — $ 15x^2 + 14x - 90 $ might be correct, but not factorable.

But let’s try factoring as $ (5x + a)(3x + b) $

Try $ (5x - 10)(3x + 9) = 15x^2 +45x -30x -90 = 15x^2 +15x -90 $

Too much.

Try $ (5x - 15)(3x + 6) = 15x^2 +30x -45x -90 = 15x^2 -15x -90 $

No.

Try $ (5x + 15)(3x - 6) = 15x^2 -30x +45x -90 = 15x^2 +15x -90 $

Same.

Try $ (15x - 18)(x + 5) = 15x^2 +75x -18x -90 = 15x^2 +57x -90 $

No.

Wait — perhaps it's $ 15x^2 + 14x - 90 $ → maybe typo in problem?

Alternatively, maybe it's $ 15x^2 + 14x - 9 $ or $ 15x^2 + 14x - 10 $

But as given, it seems not factorable.

Wait — let’s try $ (15x - 18)(x + 5) $ → already did.

Wait — what if $ (5x + 18)(3x - 5) = 15x^2 -25x +54x -90 = 15x^2 +29x -90 $

No.

Wait — try $ (3x + 10)(5x - 9) = 15x^2 -27x +50x -90 = 15x^2 +23x -90 $

No.

Wait — try $ (3x - 10)(5x + 9) = 15x^2 +27x -50x -90 = 15x^2 -23x -90 $

No.

Try $ (15x + 18)(x - 5) = 15x^2 -75x +18x -90 = 15x^2 -57x -90 $

No.

I think this one is not factorable over integers.

But let’s assume it's a typo and suppose it was meant to be $ 15x^2 + 14x - 9 $ or $ 15x^2 + 14x - 10 $

But as written, no integer factors.

Alternatively, maybe it's $ 15x^2 + 14x - 90 $, and we can factor out GCF? GCF of 15,14,90 is 1.

So final conclusion: This expression does not factor over integers.

But since it's on a worksheet, maybe I missed something.

Wait — try $ (15x - 18)(x + 5) $ → no

Wait — try $ (5x - 6)(3x + 15) = 15x^2 +75x -18x -90 = 15x^2 +57x -90 $

No.

Wait — try $ (15x + 10)(x - 9) = 15x^2 -135x +10x -90 = 15x^2 -125x -90 $

No.

I think it's safe to say #1 cannot be factored over integers.

But let’s double-check with calculator or alternate method.

Alternatively, maybe it's $ 15x^2 + 14x - 90 $, and the answer is $ (5x - 10)(3x + 9) $? No, that gives $ 15x^2 +15x -90 $

Close.

Wait — maybe the original problem has a typo.

Given that all others work, perhaps #1 is $ 15x^2 + 14x - 9 $ or $ 15x^2 + 14x - 10 $

But as written, not factorable.

However, let’s try one more idea.

Try $ (15x - 18)(x + 5) $: $ 15x^2 +75x -18x -90 = 15x^2 +57x -90 $

No.

Wait — try $ (5x - 10)(3x + 9) $: $ 15x^2 +45x -30x -90 = 15x^2 +15x -90 $

Still not.

Wait — try $ (5x + 15)(3x - 6) = 15x^2 -30x +45x -90 = 15x^2 +15x -90 $

Same.

No.

So after careful analysis, #1 is not factorable over integers.

But perhaps the intended answer is:

Wait — let’s try $ (5x - 10)(3x + 9) $ → no.

Wait — maybe it's $ 15x^2 + 14x - 90 $, and they want us to factor as:

Wait — try $ (3x + 10)(5x - 9) = 15x^2 -27x +50x -90 = 15x^2 +23x -90 $

No.

I think it's best to conclude that this expression does not factor nicely, possibly due to a typo.

But let’s assume the problem is correct and see if there’s any way.

Wait — try $ (15x - 18)(x + 5) $: no

Or $ (15x + 18)(x - 5) $: $ 15x^2 -75x +18x -90 = 15x^2 -57x -90 $

No.

After extensive checking, I believe #1 is not factorable over integers.

But let’s move on.

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Final Answers:

1. $ 15x^2 + 14x - 90 $ → Not factorable over integers (possibly typo)

2. $ (8x - 7)(2x - 5) $

3. $ 3(2x - 1)(4x + 5) $

4. $ (2x + 9)(x + 9) $

5. $ (4x + 1)(2x - 1) $

6. $ 3(x + 3)(x - 1) $

7. $ 3(2x + 1)(5x - 7) $

8. $ 2(5x + 2)(4x + 11) $

9. $ (5x + 6)(x + 1) $

10. $ (7x + 5)(x - 2) $

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Possible Correction for #1:

If the expression was $ 15x^2 + 14x - 9 $, then:

$ ac = -135 $, need two numbers that multiply to $-135$, add to $14$

Try: $ 15 $ and $-9 $: $ 15 - 9 = 6 $

$ 27 $ and $-5 $: $ 27 - 5 = 22 $

$ 18 $ and $-7.5 $: no

$ 10 $ and $-13.5 $: no

No.

Try $ 15x^2 + 14x - 10 $: $ ac = -150 $, need add to 14

Try $ 20 $ and $-7.5 $: no

$ 25 $ and $-6 $: $ 25 - 6 = 19 $

$ 15 $ and $-10 $: $ 15 - 10 = 5 $

$ 30 $ and $-5 $: $ 30 - 5 = 25 $

No.

Try $ 15x^2 + 14x - 90 $ — maybe it's acceptable to write "prime" or "does not factor"

But since others work, likely typo.

Perhaps it's $ 15x^2 + 14x - 90 $, and we are to leave it.

But based on standard problems, it's likely not factorable.

So final answers:

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✔ Final Answer Key:

1. Not factorable over integers (or possibly typo)

2. $ (8x - 7)(2x - 5) $

3. $ 3(2x - 1)(4x + 5) $

4. $ (2x + 9)(x + 9) $

5. $ (4x + 1)(2x - 1) $

6. $ 3(x + 3)(x - 1) $

7. $ 3(2x + 1)(5x - 7) $

8. $ 2(5x + 2)(4x + 11) $

9. $ (5x + 6)(x + 1) $

10. $ (7x + 5)(x - 2) $

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Let me know if you'd like to verify #1 with decimals or if you suspect a typo.