USE THE METHOD OF DIRECT TO PROVE: If a is an odd integer, then a’2 + 3a +5 is odd.

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Thus $a = 2c +1$ for some integer $c,$ by definition of an odd number. Then

$a^2+3a+5=(2c+1)^2+3(2c+1)+5$

$=4c^2+4c+1+6c+3+5=4c^2+10c+9$

$=2(2c^2+5c+4)+1=2b+1,$

where $b=2c^2+5c+4.$

Therefore by definition of an odd number, if $a$ is an odd integer, then $a^2 + 3a +5$ is odd.