# To multiply: the given binomial using FOIL method. \left(\frac{

To multiply: the given binomial using FOIL method.
$$\displaystyle{\left({\frac{{{2}}}{{{3}}}}{a}-{b}^{{{2}}}\right)}{\left({\frac{{{2}}}{{{3}}}}{a}^{{{2}}}+{b}^{{{2}}}\right)}$$

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Sevensis1977
Step 1
FOIL Method: it is a mnemonic for the standard method of multiplying of two binomials.
F stands for the product of first terms.
O stands for the product of outer terms.
I stand for the product of inner terms.
L stands for the product of last terms.
The given equation is
$$\displaystyle{\left({\frac{{{2}}}{{{3}}}}{a}-{b}^{{{2}}}\right)}{\left({\frac{{{2}}}{{{3}}}}{a}^{{{2}}}+{b}^{{{2}}}\right)}$$
Now, multiply the given binomial usinf FOIL method,
$$\displaystyle{\left({\frac{{{2}}}{{{3}}}}{a}-{b}^{{{2}}}\right)}{\left({\frac{{{2}}}{{{3}}}}{a}+{b}^{{{2}}}\right)}$$
$$\displaystyle={\left({\frac{{{2}}}{{{3}}}}{a}\right)}{\left({\frac{{{2}}}{{{3}}}}{a}\right)}+{\left({\frac{{{2}}}{{{3}}}}{a}\right)}{\left({b}^{{{2}}}\right)}+{\left(-{b}^{{{2}}}\right)}{\left({\frac{{{2}}}{{{3}}}}{a}\right)}+{\left({b}^{{{2}}}\right)}{\left(-{b}^{{{2}}}\right)}$$
$$\displaystyle{\left({\frac{{{2}}}{{{3}}}}{a}-{b}^{{{2}}}\right)}{\left({\frac{{{2}}}{{{3}}}}{a}^{{{2}}}+{b}^{{{2}}}\right)}$$
$$\displaystyle={\frac{{{4}}}{{{9}}}}{a}^{{{2}}}+{\frac{{{2}}}{{{3}}}}{a}{b}^{{{2}}}-{\frac{{{2}}}{{{3}}}}{a}{b}^{{{2}}}-{b}^{{{4}}}$$
$$\displaystyle{\left({\frac{{{2}}}{{{3}}}}{a}-{b}^{{{2}}}\right)}{\left({\frac{{{2}}}{{{3}}}}{a}^{{{2}}}+{b}^{{{2}}}\right)}={\frac{{{4}}}{{{9}}}}{a}^{{{2}}}-{b}^{{{4}}}$$
Hence, the multiplication of the given binomial is
$$\displaystyle{\left({\frac{{{2}}}{{{3}}}}{a}-{b}^{{{2}}}\right)}{\left({\frac{{{2}}}{{{3}}}}{a}^{{{2}}}+{b}^{{{2}}}\right)}={\frac{{{4}}}{{{9}}}}{a}^{{{2}}}-{b}^{{{4}}}$$