# Write the final factorization for each problem. 12a^3 + 20a^2b - 9ab^2 - 15b^3

Question
Polynomial factorization
Write the final factorization for each problem.
$$\displaystyle{12}{a}^{{3}}+{20}{a}^{{2}}{b}-{9}{a}{b}^{{2}}-{15}{b}^{{3}}$$

2021-02-26
Step 1
GCF of $$\displaystyle{12}{a}^{{3}}{\quad\text{and}\quad}{20}{a}^{{2}}{b}{i}{s}={4}{a}^{{2}}$$
GCF of $$\displaystyle-{9}{a}{b}^{{2}}{\quad\text{and}\quad}-{15}{b}^{{3}}{i}{s}=-{3}{b}^{{2}}$$
Factor out $$\displaystyle{4}{a}^{{2}}$$ from the first two terms and then factor out $$\displaystyle-{3}{b}^{{2}}$$ from the last two terms.
$$\displaystyle{12}{a}^{{3}}+{20}{a}^{{2}}{b}-{9}{a}{b}^{{2}}-{15}{b}^{{3}}$$
$$\displaystyle={4}{a}^{{2}}{\left({3}{a}+{5}{b}\right)}-{3}{b}^{{2}}{\left({3}{a}+{5}{b}\right)}$$
Step 2
Then we can factor out (3a+5b) from both terms.
$$\displaystyle{4}{a}^{{2}}{\left({3}{a}+{5}{b}\right)}-{3}{b}^{{2}}{\left({3}{a}+{5}{b}\right)}$$
$$\displaystyle={\left({3}{a}+{5}{b}\right)}{\left({4}{a}^{{2}}-{3}{b}^{{2}}\right)}$$
Result:$$\displaystyle{\left({3}{a}+{5}{b}\right)}{\left({4}{a}^{{2}}-{3}{b}^{{2}}\right)}$$

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