Given:
Polynomials: [tex]a+3 \text { and }-2 a^{2}+15 a+6 b^{2}[/tex]
To find:
The product of the polynomials.
Solution:
[tex](a+3)(-2 a^{2}+15 a+6 b^{2})[/tex]
Using distributive property: [tex]x(y+z)=xy+xz[/tex]
[tex](a+3)(-2 a^{2}+15 a+6 b^{2})=a(-2 a^{2}+15 a+6 b^{2})+3(-2 a^{2}+15 a+6 b^{2})[/tex]
Now multiply each of the first term with each of the second term.
[tex]=a\left(-2 a^{2}\right)+a \cdot 15 a+a \cdot 6 b^{2}+3\left(-2 a^{2}\right)+3 \cdot 15 a+3 \cdot 6 b^{2}[/tex]
Applying plus minus rule: [tex]+(-x)=-x[/tex]
[tex]=-2 a^{2} \cdot a+15 a \cdot a+6 a\cdot b^{2}-3 \cdot 2 a^{2}+3 \cdot 15 a+3 \cdot 6 b^{2}[/tex]
Apply the exponent rule: [tex]x^{n} \cdot x^{m}=x^{n+m}[/tex]
[tex]=-2 a^{3}+15 a^2+6 a b^{2}-6 a^{2}+45 a+18 b^{2}[/tex]
Add or subtract the like terms:
[tex]=-2 a^{3}+15 a^2-6a^2+6 a b^{2}+45 a+18 b^{2}[/tex]
[tex]=-2 a^{3}+9 a^{2}+6 a b^{2}+45 a+18 b^{2}[/tex]
Arrange in the order.
[tex]=-2 a^{3}+9 a^{2}+45 a+6 a b^{2}+18 b^{2}[/tex]
The product of [tex]a+3 \text { and }-2 a^{2}+15 a+6 b^{2}[/tex] [tex]-2 a^{3}+9 a^{2}+45 a+6 a b^{2}+18 b^{2}[/tex].
