https://file.aiquickdraw.com/mathbot/file/1727302785021vxuasr16.jpg

Let's go through the two problems step by step.

### Problem (a):
We are given \( a(x) = x^2 + 2x - 5 \) and asked to find \( a(x+3) \).

To do this, we substitute \( x+3 \) into the expression for \( a(x) \).

\[
a(x+3) = (x+3)^2 + 2(x+3) - 5
\]

Now, expand and simplify:

\[
(x+3)^2 = x^2 + 6x + 9
\]
\[
2(x+3) = 2x + 6
\]
Substitute these back into the expression:

\[
a(x+3) = (x^2 + 6x + 9) + (2x + 6) - 5
\]

Simplify:

\[
a(x+3) = x^2 + 6x + 9 + 2x + 6 - 5
\]
\[
a(x+3) = x^2 + 8x + 10
\]

### Problem (b):
We are given \( b(x) = 5x + 3 \) and asked to find \( b(x^2 - 4) \).

To solve, we substitute \( x^2 - 4 \) into the expression for \( b(x) \).

\[
b(x^2 - 4) = 5(x^2 - 4) + 3
\]

Now, expand:

\[
b(x^2 - 4) = 5x^2 - 20 + 3
\]
\[
b(x^2 - 4) = 5x^2 - 17
\]

### Final Answers:
- (a) \( a(x+3) = x^2 + 8x + 10 \)
- (b) \( b(x^2 - 4) = 5x^2 - 17 \)

Would you like more details on these solutions or have any questions? Here are some related questions you could explore:

1. What would be the result if \( a(x) = 3x^2 + 4x - 2 \) instead?
2. How would you solve \( a(2x+1) \) for the same function \( a(x) \)?
3. What is the general method for evaluating \( a(f(x)) \) for any function \( f(x) \)?
4. How can this process be applied to more complex functions, such as logarithmic or trigonometric functions?
5. If \( b(x) \) were a quadratic function, how would you approach \( b(x^2 - 4) \)?

**Tip:** When substituting into a function, always expand carefully and combine like terms to avoid errors.

Evaluate the following: a) Given a(x) = x^2 + 2x - 5, find a(x+3); b) Given b(x) = 5x + 3, find b(x^2 - 4)

Learn how to evaluate function substitutions for algebraic expressions. This problem involves finding a(x+3) and b(x^2-4) based on the given functions a(x) = x^2 + 2x - 5 and b(x) = 5x + 3. Suitable for Grades 9-10.

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