Question - Expanding and Simplifying Polynomial Expressions

Simplify the given expressions:

\textbf{For the 2/ expression:}

(x + 1)(x + 2)(x + 3)(x + 4) - 24

=(x^2 + 5x+ 6)(x^2 + 7x + 12) - 24

=(x^4 + 12x^3 + 46x^2 + 72x + 24) - 24

=x^4 + 12x^3 + 46x^2 + 72x

\textbf{For the 4/ expression:}

(x^2 + x)^2 + 4x^2 + 4x -12

=(x^4 + 2x^3 + x^2) + 4x^2 + 4x -12

=x^4 + 2x^3 + 5x^2 + 4x -12

\textbf{For the 6/ expression:}

(x + a)(x + 2a)(x + 3a)(x + 4a) + a^4

=(x^2 + 3ax + 2a^2)(x^2 + 7ax + 12a^2) + a^4

=(x^4 + 10ax^3 + (3*12+7*2)a^2x^2 + (3*7+2*12)ax^3 + 24a^4) + a^4

=x^4 + 10ax^3 + 38a^2x^2 + 33a^3x + 25a^4

\textbf{For the 8/ expression:}

(x^2 + x)^2 + 3(x^2 + x) + 2

=(x^4 + 2x^3 + x^2) + 3x^2 + 3x + 2

=x^4 + 2x^3 + 4x^2 + 3x + 2

\textbf{For the 10/ expression:}

(x^2 + 2x)^2 + 9x^2 + 18x + 20

=(x^4 + 4x^3 + 4x^2) + 9x^2 + 18x + 20

=x^4 + 4x^3 + 13x^2 + 18x + 20

\textbf{For the 12/ expression:}

(x + 2)(x + 4)(x + 6)(x + 8) + 16

=(x^2 + 6x + 8)(x^2 + 14x + 48) + 16

=(x^4 + 20x^3 + 144x^2 + 384x + 128) + 16

=x^4 + 20x^3 + 144x^2 + 384x + 144