questions

Deductive Proof &amp; Proof by Exhaustion

Let <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>n be an integer.
Prove that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>n</mi><mo>+</mo><mn>8</mn></mrow><annotation encoding="application/x-tex">n^2+5n+8</annotation></semantics></math>n2+5n+8
is even.

Writes <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>n</mi><mo>+</mo><mn>8</mn><mo>=</mo><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>5</mn><mo stretchy="false">)</mo><mo>+</mo><mn>8</mn></mrow><annotation encoding="application/x-tex">n^2+5n+8=n(n+5)+8</annotation></semantics></math>n2+5n+8=n(n+5)+8 and establishes that <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">n(n+5)</annotation></semantics></math>n(n+5) is even because <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>n and <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>+</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">n+5</annotation></semantics></math>n+5 have opposite parity.

Concludes that <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>n</mi><mo>+</mo><mn>8</mn></mrow><annotation encoding="application/x-tex">n^2+5n+8</annotation></semantics></math>n2+5n+8 is even because it is the sum of two even integers.