Sure, let's simplify the given expression step-by-step.
Expression to simplify:
[tex]\[ y^3 + 4x(6y^3 + 3x) \][/tex]
Step 1: Distribute [tex]\(4x\)[/tex] inside the parentheses.
[tex]\[ 4x(6y^3 + 3x) = 4x \cdot 6y^3 + 4x \cdot 3x \][/tex]
Step 2: Perform the multiplications.
[tex]\[ 4x \cdot 6y^3 = 24xy^3 \][/tex]
[tex]\[ 4x \cdot 3x = 12x^2 \][/tex]
Now, adding these results:
[tex]\[ 4x(6y^3 + 3x) = 24xy^3 + 12x^2 \][/tex]
Step 3: Combine all terms.
[tex]\[ y^3 + 24xy^3 + 12x^2 \][/tex]
Step 4: Factor out common factors where possible. In this case, factor out a common term of [tex]\(12x\)[/tex] from the parts of the expression that can be factored.
[tex]\[ y^3 + 24xy^3 + 12x^2 = y^3 + 12x (2y^3 + x) \][/tex]
So, the simplified form of the expression is:
[tex]\[ y^3 + 12x (x + 2y^3) \][/tex]
Therefore:
[tex]\[ y^3 + 4x(6y^3 + 3x) = 12x (x + 2y^3) + y^3 \][/tex]
