Expand (a+3b)²: Converting to Multiplication and Addition Forms

To solve this problem, we need to express $(a+3b)^2$ in two forms: as a multiplication of like terms and as an addition of polynomial terms.

• Step 1: Express as a multiplication
  The expression $(a+3b)^2$ can be rewritten as $(a+3b)(a+3b)$. This indicates the binomial is multiplied by itself.
• Step 2: Expand using the binomial theorem
  We apply the formula for squaring a binomial:
  $(x+y)^2 = x^2 + 2xy + y^2$.
  Here, let $x = a$ and $y = 3b$. Applying the formula we get:
  $(a+3b)^2 = a^2 + 2(a)(3b) + (3b)^2$.
• Step 3: Simplify
  Perform the calculations for each term:
  • $a^2$: The square of $a$.
  • $2(a)(3b) = 6ab$: Multiply the terms and the coefficients.
  • $(3b)^2 = 9b^2$: Square the coefficient and the variable.
  Combining these, the expanded form is:
  $a^2 + 6ab + 9b^2$.

Thus, the expression as a multiplication is $(a+3b)(a+3b)$, and as an addition, it is $a^2 + 6ab + 9b^2$.

Therefore, the solution to the problem is $(a+3b)(a+3b)$ and $a^2+6ab+9b^2$.