The equation given in the image is:

$(x + a)^3 = x^3 + px^2 + qx + 64$

We need to solve for the constants $p$ and $q$.

Step-by-Step Solution

1. Expand the Left Side:
   Let's expand the left side of the equation using the binomial expansion formula:

   $(x + a)^3 = x^3 + 3ax^2 + 3a^2x + a^3$

2. Equate Coefficients:
   Now, compare this expansion with the right side of the equation:

   $x^3 + 3ax^2 + 3a^2x + a^3 = x^3 + px^2 + qx + 64$

   Equating the coefficients of like terms on both sides:

   • Coefficient of $x^2$: $3a = p$
   • Coefficient of $x$: $3a^2 = q$
   • Constant term: $a^3 = 64$

3. Solve for $a$, $p$, and $q$:

   • From $a^3 = 64$, we find:

     $a = \sqrt[3]{64} = 4$

   • Substitute $a = 4$ into the expressions for $p$ and $q$:

     $p = 3a = 3 \times 4 = 12$

     $q = 3a^2 = 3 \times 4^2 = 3 \times 16 = 48$

Solution:

• $p = 12$
• $q = 48$

Would you like more details on the binomial expansion or have any questions? Here are some related questions:

1. How does the binomial theorem work for higher powers?
2. What are some other ways to expand polynomials?
3. What if we had a different cubic equation to compare with?
4. How do we solve for constants in equations involving polynomials?
5. What if the constants are complex numbers?

Tip: Always equate coefficients of like terms when comparing polynomials to solve for unknowns efficiently.