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1. The first equation seems to be:  
   \[ (x + 3)(x + 7) = \]

2. The second equation appears to be:  
   \[ (x + 5)(x + 1)(x + 4)(x + 9)(x + 6)(x + 2) = \]

Let me solve each of these step by step.

### 1. Expanding \((x + 3)(x + 7)\):

\[
(x + 3)(x + 7) = x(x + 7) + 3(x + 7)
\]

Expanding each term:

\[
= x^2 + 7x + 3x + 21
\]

Combining like terms:

\[
= x^2 + 10x + 21
\]

### 2. Expanding \((x + 5)(x + 1)(x + 4)(x + 9)(x + 6)(x + 2)\):

This expansion is more complex and involves multiplying each pair step-by-step. We'll start by expanding pairs of expressions:

- First pair: \((x + 5)(x + 1)\)
\[
= x^2 + x + 5x + 5 = x^2 + 6x + 5
\]

- Second pair: \((x + 4)(x + 9)\)
\[
= x^2 + 9x + 4x + 36 = x^2 + 13x + 36
\]

- Third pair: \((x + 6)(x + 2)\)
\[
= x^2 + 2x + 6x + 12 = x^2 + 8x + 12
\]

Now, we multiply these results step by step:

- First multiply \((x^2 + 6x + 5)\) by \((x^2 + 13x + 36)\):

\[
(x^2 + 6x + 5)(x^2 + 13x + 36)
\]

Expanding each term:

\[
= x^4 + 13x^3 + 36x^2 + 6x^3 + 78x^2 + 216x + 5x^2 + 65x + 180
\]

Combining like terms:

\[
= x^4 + 19x^3 + 119x^2 + 281x + 180
\]

- Now multiply the above result by \((x^2 + 8x + 12)\):

\[
(x^4 + 19x^3 + 119x^2 + 281x + 180)(x^2 + 8x + 12)
\]

Expanding:

\[
= x^6 + 8x^5 + 12x^4 + 19x^5 + 152x^4 + 228x^3 + 119x^4 + 952x^3 + 1428x^2 + 281x^3 + 2248x^2 + 3372x + 180x^2 + 1440x + 2160
\]

Combining like terms:

\[
= x^6 + 27x^5 + 283x^4 + 1451x^3 + 3856x^2 + 4812x + 2160
\]

### Final Results:

1. \((x + 3)(x + 7) = x^2 + 10x + 21\)

2. \((x + 5)(x + 1)(x + 4)(x + 9)(x + 6)(x + 2) = x^6 + 27x^5 + 283x^4 + 1451x^3 + 3856x^2 + 4812x + 2160\)

Would you like any further details or explanations? Here are some related questions:

1. How do you expand binomials using the distributive property?
2. What are the steps to multiply polynomials of higher degrees?
3. How can you factorize polynomials after expansion?
4. What are the applications of polynomial multiplication in real-world problems?
5. Can you demonstrate how to verify the results of polynomial expansion?

**Tip:** When expanding polynomials, always remember to group like terms together at each step to simplify the expression effectively.

(x + 3)(x + 7) = ?; (x + 5)(x + 1)(x + 4)(x + 9)(x + 6)(x + 2) = ?

Learn how to expand and multiply polynomials such as (x + 3)(x + 7) and the higher degree polynomial (x + 5)(x + 1)(x + 4)(x + 9)(x + 6)(x + 2) with a detailed step-by-step solution. This problem covers key algebraic concepts including the distributive property and multiplication of polynomials, suitable for students in Grades 10-12.

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