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An integer is called even if it can be divided by 2 without leaving a remainder. In other words, if an integer m is even, it can be expressed in the form m = 2k, where k is another integer. For example, 4 is even because it can be written as 2 x 2, showing that it is divisible by 2.

Conversely, an integer is called odd if it leaves a remainder of 1 when divided by 2. An odd integer m can therefore be written as m = 2k + 1, with k again being an integer. Take 5, for instance, which is odd because 5 = 2 x 2 + 1.

The difference between even and odd integers is foundational to understanding their behavior in equations and proving related mathematical statements. For example, the sum of any two even numbers is even, while the sum of two odd numbers is also even. However, the sum of an even number and an odd number is odd, and that is key to our proofs.

Proof by substitution is a method used to show that an expression holds true under certain conditions. This involves taking a known variable condition, such as an integer being even or odd, and substituting the algebraic form that represents that condition into an equation. Through this method, a broader statement can be validated for all the integers that satisfy the initial condition.

For instance, in our exercise, the statement (a) assumes m is an even integer. By substituting m with 2k, where k is an integer, we use the property that defines even numbers to test the validity of the given statement. This is a powerful tool because, if successful, it proves that the initial claim is always true for every even or odd integer, offering a universal understanding of that mathematical relationship.

Using proof by substitution is effective in a wide range of mathematical areas, including algebra and number theory. It provides a clear means to verify algebraic identities, inequalities, and various theoretical propositions.

Algebraic manipulation involves rearranging, combining, and simplifying algebraic expressions to reveal their properties or to prove certain statements about them. In solving our exercise, algebraic manipulation is the act of expanding, factoring, and reordering terms to take advantage of the known properties of even and odd integers.

When we expanded the expression for the even case, 3(2k)^2 + 2(2k) + 3, and factored out the 2, we made the underlying property of evenness more apparent. The even part of the expression was distinguished as multiply by 2, allowing us to focus on the last term, +3, to indicate oddness. Similarly, for the odd case, simplifying 3(2k+1)^2 + 7(2k+1) + 12 and factoring out the 2 made it clear that the result would be even.

Good algebraic manipulation hinges on knowing these properties and combining like terms, factoring common factors, and employing distributive, associative, and commutative properties. It is a vital skill in mathematics that enables students to streamline expressions and solve complex problems. Mastering this skill can make the difference between understanding and merely following steps in mathematical problem solving.