Given the rational expression;
[tex]\frac{16b^2+40b+25}{4b+5}[/tex]
We shall begin by factorizing the numerator as follows;
[tex]\begin{gathered} 16b^2+40b+25 \\ \text{Note that the coefficient of b}^2\text{ is greater than 1} \\ \text{Therefore we shall multiply the constant by the coefficient of b}^2 \\ \text{That gives us;} \\ 16\times25=400 \\ We\text{ shall now use the sum-product method, which is;} \\ \text{The factors of the constant 400} \\ S\text{hall also sum up to the coefficient of b } \\ \text{These factors are +20, +20} \\ \text{Therefore;} \\ 16b^2+40b+25\text{ becomes;} \\ 16b^2+20b+20b+25 \\ \text{Factorize by groups of two and we'll have} \\ 4b(4b+5)+5(4b+5) \\ \text{This becomes;} \\ (4b+5)(4b+5) \end{gathered}[/tex]
The rational expression now becomes;
[tex]\frac{(4b+5)(4b+5)}{(4b+5)}[/tex]
