How do you factor `15x^2 + 4x - 4 = 0`?

It is a matter of spotting possible combinations and trying them out. They can be tried out on paper or in your head.

Factors of 15 `->`{1,15} ; {3,5}

Factors of 4 `->`{1,4} ; {2,2}

Cross link factors of 4 with factors of 15

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Consider: {1,4} linked to {1,15}

`(1xx15)-(4xx1)=11" So this is no good for obtaining "4x`
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Consider: {2,2} linked to {1,15}

`(2xx15)-(2xx1)=28" So this is no good for obtaining "4x`
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Consider: {1,4} linked to {3,5}

`(3xx4)-(1xx5)=7" So this is no good for obtaining "4x`
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Consider: {2,2} linked to {3,5}

`(color(green)(2xx5))-(color(red)(2xx3)) = 4color(brown)(" this is the combination we need")`
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`(color(red)(3)x color(green)( +2))(color(green)(5)x color(red)(- 2))`